Reconstruction problems of convex bodies from surface area measures and lightness functions

Abstract

First, we build a computational procedure to reconstruct the convex body from a pre-given surface area measure. Nontrivially, we prove the convergence of this procedure. Then, the sufficient and necessary conditions of a Sobolev binary function to be a lightness function of a convex body are investigated. Finally, we present a computational procedure to compute the curvature function from a pre-given lightness function, and then we reconstruct the convex body from this curvature function by using the first procedure. The convergence is also proved. The main computations in both procedures are simply about the spherical harmonics.

1 Introduction

Reconstruction problems of convex bodies (compact convex sets with nonempty interiors in $\mathbb {R}^n$ ) for some known geometric functions and measures (e.g., area measures, curvatures, brightness function, and lightness functions in this paper) have caught great attention in convex geometric analysis. For instance, a convex body K can be uniquely determined, up to translations, by its surface area measure $S(K,\cdot )$ , which is a spherical Borel measure defined by

$$ \begin{align*}S(K,\eta) = \mathcal{H}^{n-1}\big(\nu_K^{-1}(\eta)\big), \quad \forall\mathrm{~Borel~set~}\eta\subset\mathbb{S}^{n-1},\end{align*} $$

where $\mathcal {H}^{n-1}$ denotes the $(n-1)$ -dimensional Hausdorff measure, and $\nu _K: \partial K \to \mathbb {S}^{n-1}$ is the Gauss map defined $\mathcal {H}^{n-1}$ -almost everywhere on the boundary $\partial K.$ Investigation about whether a spherical Borel measure $\mu $ is the surface area measure of a convex body K is the classical Minkowski problem. Its complete solution goes back to the work of Aleksandrov and Fenchel-Jessen (see [1, 19, 28] for references). Around 1937, Aleksandrov also showed that a centrally symmetric convex body K is uniquely determined, up to translations, by its brightness function

$$\begin{align*}b_K(v) = \frac12\int_{\mathbb{S}^{n-1}} |u\cdot v| dS(K,u).\end{align*}$$

Independently, Campi [8] and Bourgain and Lindenstrauss [6] proved a stability version of this result.

1.1 Reconstruction problems for a geometric function (or measure)

In view of much of previous work, we roughly summarize the reconstruction problems into the following three types:

1. (I) Study whether a geometric function (or measure) can uniquely and stably determine a convex body.

2. (II) Given a function (or measure) $\Xi $ , study both the necessary and sufficient conditions of $\Xi $ so that there is a convex body K whose geometric function (or measure) is the pre-given $\Xi $ .

3. (III) Suppose the geometric function (or measure) of a convex body K is given. Try to build a computational procedure to reconstruct the related convex body; prove the convergence; and design an algorithm based on the computational procedure, and then implement it and test it.

Aleksandrov’s projection theorem and its stability versions belong to the first topic (I). The spherical harmonic is the powerful tool therein. This technique can also solve problem (II) for brightness function (see, e.g., Lemma 4.4). Gardner and Milanfar [13, 14] first studied the type (III) problem for brightness function. In fact, they provided algorithms to reconstruct the origin-symmetric convex body from its given brightness function. Before that, they gave an algorithm to reconstruct the polytope from the pre-given discrete measure. Gardner, Kiderlen, Milanfar, and Peyman [12] studied the convergence of related algorithms, which belongs to the topic (III). Bianchi, Gardner, and Kiderlen [2] studied problem (III) for covariograms of convex bodies. For studies about the reconstruction problems for other geometric functions (or measures), see, e.g., [20, 21, 23, 26].

The Minkowski problems for geometric measures always involve type (I) and type (II). Except for the surface area measure, Aleksandrov, Fenchel, and Jessen also introduced the mixed area measures, and related problems are called the Christoffel–Minkowski problem. For these geometric measures, type (I) problems are completely solved, but type (II) problems are not (see [17] for a regular case). While the curvature measures were introduced by Federer about half a century ago, its dual counterpart was introduced quite recently by Huang, Lutwak, Yang, and Zhang [19, 24]. The studies of the reconstruction problems of type (II) for the new geometric measures have caught great attention, in both convex geometry and partial differential equations (see, e.g., [4, 3, 10, 22, 30]). However, the problem of type (I) for the dual curvature measure is only solved in one special case on the plane [5]. It seems that the problem of type (III) is only considered for surface area measure. For instance, see [14, 23] for discrete case in $\mathbb {R}^n$ , and see [20] for smooth case in $\mathbb {R}^3$ . In [14], the authors reconstructed polytopes from pre-given discrete measures, and they proved the convergence of their algorithm; in [20], by using the spherical harmonic, the authors gave an effective numerical algorithm for smooth curvature functions in $\mathbb {R}^3$ , but they did not rigorously prove the convergence.

Given a convex body K in $\mathbb {R}^n$ , a lightness function [9] is a binary function that gives the total reflected light resulting from illumination by a light source at infinity in the direction w that is visible when looking in the direction v. An interesting model appeared in [9] is the Lambertian lightness function, which means that it follows Lambert’s cosine law, and it is formally defined by

(1.1) $$ \begin{align} L_K(v,w) =\int_{\mathbb{S}^{n-1}} (v\cdot u)_-(w\cdot u)_+dS(K,u), \end{align} $$

where $(t)_+$ and $(t)_-$ denote, respectively, the positive part and negative part of a real number t. In view of the general description of lightness function, the authors [9] also introduced a more general lightness function

(1.2) $$ \begin{align}Q_K(v,w) = \int_{\mathbb{S}^{n-1}} (v\cdot u)_- f_w(u) dS(K,u),\end{align} $$

where $f_w:\mathbb {S}^{n-1} \to [0,\infty )$ is positive and continuous in the open half-space $w^+ = \{x: x\cdot w>0\}$ and vanishes in its complement. This assumption is to fulfill that a watcher from the viewing direction w can only see a visible part of $\partial K$ . The authors [9] gave exhaustive studies about the first topic (I), for these lightness functions.

The aim of this is to study (the theoretical part of) problem (III) for surface area measure, and problems (II) and (III) for lightness functions. We explicitly restate them as follows:

• • (III) Given the surface area measure $S(K,\cdot )$ , try to build a computational procedure to reconstruct the convex body, and prove the convergence.

• • (II) What are the sufficient and necessary conditions of a binary function $G:\mathbb {S}^{n-1}\times \mathbb {S}^{n-1}\to \mathbb {R}$ so that there is a convex body K whose lightness function $Q_K$ is the pre-given G?

• • (III) Given a lightness function $Q_K$ , try to build a computational procedure to reconstruct the convex body, and prove the convergence.

The spherical harmonics will be an important tool in this paper. They are the eigenvectors of spherical Laplace–Beltrami operator $\Delta _S:C^\infty (\mathbb {S}^{n-1})\to C^\infty (\mathbb {S}^{n-1})$ . For more detailed introduction, see Section 2.5. We also refer to [6, 15, 27–29] as good general references.

In Section 3, we give a computational procedure to reconstruct the convex body from a pre-given surface area measure $\mu $ on $\mathbb {S}^{n-1}$ . The main computations in the procedures will be the integrals of spherical harmonics and the mixed discriminants (which will be introduced in Section 2.3) of their Hessians. First, we consider a sequence of convex optimization problems ( $P_N$ ), and we prove in Theorem 3.5 that the solutions of ( $P_N$ ) will converge to a dilation of the convex body whose surface area measure is the pre-given $\mu $ . Noticing that ( $P_N$ ) has infinite many constraints, it is very difficult to solve ( $P_N$ ) directly. Therefore, we construct the subproblem ( $P_{N,k}$ ), which is an approximation problem of ( $P_N$ ). Nontrivially, we prove in Theorem 3.3 that ( $P_{N,k}$ ) are also convex optimization problems when k is sufficiently large, but with only finitely many constraints. Since problem ( $P_{N,k}$ ) are convex optimization problems with only finite many constraints, they can be efficiently solved by many existing optimization package, and we do not build the explicit algorithm. Finally, by combining Theorem 3.3 with Theorem 3.5, we conclude that the solution of ( $P_{N,k}$ ) (after normalization) converges (in Hausdorff metric) to the convex body K whose surface area measure is $\mu $ .

In Section 4.2, using the language of spherical harmonics, we give both the necessary and sufficient conditions of a binary function to be a lightness function, under the Sobolev setting. We use $H_k(\mathbb {S}^{n-1})$ to denote the Sobolev space of order k, and the detailed introduction can be seen in Section 2.4. We take the Lambertian lightness function $L_K(v,w)$ as an example, since it has more physical explanations. A trivial attempt is that if K is smooth and symmetric,

$$\begin{align*}L_K(v,-v) = \frac12\int_{\mathbb{S}^{n-1}} |v\cdot u|^{2} dS(K,u),\end{align*}$$

which is the $L^2$ cosine transform of $S(K,\cdot )$ . However, it is well known that the $L^p$ cosine transform is not surjective from $C_e^\infty (\mathbb {S}^{n-1})$ (even functions in $C^\infty (\mathbb {S}^{n-1})$ ) into $C_e^\infty (\mathbb {S}^{n-1})$ when p is even.

In view of this, the interesting but also reasonable thing in Section 4 is that the convex body K can be determined by the even expansion of the binary function $L_K$ without symmetric assumption, which is different from the brightness function. Moreover, the representations in the main theorems naturally lead to an algorithm to compute the curvature function of a convex body K. Our proof relies on some results that we slightly develop in Section 4.1, including the eigenvalues and the reverse of the partial cosine transform

$$\begin{align*}\mathcal{C}_- f (v) = \int_{\mathbb{S}^{n-1}}(v\cdot u)_-f(u)du\end{align*}$$

in the Sobolev setting. We note that the Sobolev setting and the tool of spherical harmonics are very suitable and convenient to build algorithms to reconstruct convex bodies from known surface area measures or lightness functions.

In Section 5, we use the main results in Section 4.2 to construct a computational procedure to compute the surface area measure from a pre-given lightness function. Then, using the Minkowski procedure (Mink-Pro) in Section 3, we are able to reconstruct the convex body K (without the symmetric assumption). The convergence of the procedure is also shown.

The main computations in the two procedures of Sections 3 and 5 are merely about the spherical harmonics $Y_{m,j}$ ’s, including the mixed discriminants of their Hessians, and the scalar products of them with given function (or measure). These computations are independent of the indices $N,k$ of the problem ( $P_{N,k}$ ), and hence can be computed in advance and stored. In other words, the two procedures are robust and easily implementable.

2 Preliminaries

In this section, we collect some definitions, notations, and basic facts about convex bodies, the Minkowski problem, Sobolev space on $\mathbb {S}^{n-1}$ , and the spherical harmonics. We refer to [11, 16, 18, 28, 29] as good general references.

2.1 Convex bodies

Let K be a nonempty convex compact set. Its support function $h_K:\mathbb {R}^n \to \mathbb {R}$ is defined by

$$ \begin{align*}h_K(u)=\max\{x\cdot u:x\in K\}.\end{align*} $$

Sometimes, we also view it as a spherical function by restricting it to $\mathbb {S}^{n-1}$ . The support function is always sublinear. Conversely, a sublinear function uniquely determines a nonempty convex compact set.

The Hausdorff distance between two nonempty compact convex sets $K_1,K_2$ is defined by

$$ \begin{align*}d_H(K_1,K_2) = \|h_{K_1}-h_{K_2}\|_\infty.\end{align*} $$

Given a function $f\in C(\mathbb {S}^{n-1})$ , denote $W[f]$ to be the Wullf shape determined by f:

$$\begin{align*}W[f] = \Big\{x\in\mathbb{R}^n: x\cdot u \le f(u), \forall u\in \mathbb{S}^{n-1}\Big\}.\end{align*}$$

If $W[f]$ has interiors, and $f_i\in C(\mathbb {S}^{n-1}), i=1,\ldots ,$ is a sequence of continuous functions that converges to $ f$ uniformly, then $W[f_i]\to W[f]$ in the Hausdorff metric.

We say the surface area measure $S(K,\cdot )$ has a density, if it is absolutely continuous with respect to the spherical Lebesgue measure. In this case, we denote the Radon–Nikodym derivative by

$$\begin{align*}F_K(u) = \frac{dS(K,u)}{du},\end{align*}$$

and we call $F_K$ the curvature function.

Given a function $h\in C^2(\mathbb {S}^{n-1}),$ denote $D^2h$ by

$$\begin{align*}D^2h = \nabla^2 h + hI,\end{align*}$$

where $\nabla ^2 h$ is the Hessian of h with respect to the standard metric on $\mathbb {S}^{n-1}$ , and I denotes the identity map. If one extend the h to a positive-homogeneous function on $\mathbb {R}^n$ , then $D^2h(u)$ is exactly the restriction of its Hessian in $\mathbb {R}^n$ to the tangent space of $\mathbb {S}^{n-1}$ at u.

In fact, if $\partial K$ is $C^2$ and has positive Gauss curvature everywhere, then $S(K,\cdot )$ has a density,

$$ \begin{align*}F_K(u)=\frac{dS(K,u)}{du}= \det(D^2 h_K),\end{align*} $$

which is the reciprocal Gauss curvature at $\nu _K^{-1}(u)\in \partial K.$

2.2 Minkowski problem and the Monge–Ampère equation

Solution to the classical Minkowski problem: Suppose $\mu$ is a finite Borel measure on satisfying:

(1) $\mu $ is not concentrated in a half-sphere;

(2) the centroid of $\mu $ is the origin:

$$\begin{align*}\int_{\mathbb{S}^{n-1}} u d\mu(u) = 0.\end{align*}$$

Then, there is a unique (up to translations) convex body K in $\mathbb R^n$ so that $\mu (\cdot ) =S(K, \cdot )$ . Moreover, the convex body K is homothetic to the solution of the maximization problem

$$\begin{align*}\max V(Q), \quad \mathrm{subject~to~that}~ Q~\mathrm{is~a~convex~body ~and~}\int_{\mathbb{S}^{n-1}} h_Q d\mu =1.\end{align*}$$

If the measure $\mu $ has a density function $g :\mathbb S^{n-1}\rightarrow \mathbb R$ with respect to the Lebesgue measure of the unit sphere $\mathbb {S}^{n-1}$ , the Minkowski problem is equivalent to the study of solution to the following Monge–Ampère equation on $\mathbb S^{n-1}$ :

(2.1) $$ \begin{align} \det (D^2h) = \det(\nabla^2 h +h I)=g. \end{align} $$

Let $f\in L^1(\mathbb {S}^{n-1})$ be nonnegative, satisfying $\int _{\mathbb {S}^{n-1}} f>0$ and

$$\begin{align*}\int_{\mathbb{S}^{n-1}} uf(u)du =0.\end{align*}$$

Then, by the solution of the Minkowski problem, there is always a convex body K such that

$$\begin{align*}dS(K,u) = f(u)du.\end{align*}$$

2.3 Mixed discriminant and mixed volume

Let $A_1,\ldots ,A_m$ be self-adjoint linear transforms from a k-dimensional Hilbert V to itself. Then, the following is a k-homogeneous polynomial of $t_1,\ldots ,t_m \in \mathbb {R}$ :

$$\begin{align*}\det \Big(t_1A_1+\cdots t_mA_m\Big) = \sum_{i_1=1}^{m}\ldots\sum_{i_{k}=1}^{m} \mathrm{D} (A_{i_1},\ldots,A_{i_{k}})t_{i_1}\ldots t_{i_{k}}.\end{align*}$$

Its coefficients $\mathrm {D} (A_{i_1},\ldots ,A_{i_{k}})$ , which are symmetric functions of $A_{i_1},\ldots ,A_{i_{k}}$ , are called the mixed discriminants of the linear transforms.

If we choose an orthonormal basis in V, these $A_i$ can be represented by symmetric $k\times k$ matrices, and we still denote them by $A_i$ . Denote the matrices $A_{r}=\left (a_{i j}^{r}\right )_{i, j=1}^{k}, r=1, \ldots , k$ , and the mixed discriminant $\mathrm {D} (A_1,\ldots ,A_k)$ can be computed by

(2.2) $$ \begin{align} \mathrm{D} \left(A_1, \ldots, A_{k}\right)=\frac{1}{k!} \sum_{\sigma \in P_{k}}\left|\begin{array}{ccc} a_{11}^{\sigma(1)} & \cdots & a_{1 k}^{\sigma(k)} \\ \vdots & & \vdots \\ a_{k1}^{\sigma(1)} & \cdots & a_{k k}^{\sigma(k)} \end{array}\right|, \end{align} $$

where $P_k$ is the group of all permutations of the set $\{1,\ldots ,k\}.$ Clearly, $\mathrm {D} (A_1,\ldots ,A_k)$ is linear in each argument.

Let $K_1,\ldots ,K_m$ be convex bodies, and let $\lambda _1,\ldots ,\lambda _m\ge 0$ . Then, the volume of $\lambda _1K_1+\cdots \lambda _m K_m$ is an nth homogeneous polynomial of $\lambda _1,\ldots ,\lambda _m$ ,

$$ \begin{align*}V\left(\lambda_{1} K_{1}+\cdots+\lambda_{m} K_{m}\right)=\sum_{i_{1}, \ldots, i_{n}=1} \mathrm{V} \left(K_{i_{1}}, \ldots, K_{i_{n}}\right) \lambda_{i_{1}} \cdots \lambda_{i_{n}}. \end{align*} $$

The coefficients $\mathrm {V} \left (K_{i_{1}}, \ldots , K_{i_{n}}\right )$ , which will be called the mixed volumes, are nonnegative, symmetric in the indices, and dependent only on $K_{i_{1}}, \ldots , K_{i_{n}}$ .

If $\partial K$ is $C^2$ and has positive Gauss curvature everywhere, then

$$\begin{align*}V(K) = \frac1n \int_{\mathbb{S}^{n-1}} h_K(u) \det (D^2h_K)(u) du.\end{align*}$$

In addition, if $h_K=t_1h_1+\cdots t_mh_m$ , for some $h_i\in C^2$ and $t_i\in \mathbb {R}$ , we have

(2.3) $$ \begin{align}\qquad V(K) = \sum_{i_1=1}^{m}\ldots \sum_{i_n=1}^{m} \Big(\frac1n\int_{\mathbb{S}^{n-1}} h_{i_n}(u) \mathrm{D} (D^2h_{i_1},\ldots,D^2h_{i_{n-1}})(u)du\Big) t_{i_1}\ldots t_{i_{n}}.\end{align} $$

Here, $\mathrm {D} (D^2h_{i_1},\ldots ,D^2h_{i_{n-1}})(u)$ should be understood as the mixed discriminant of linear transforms restricted on $u^\perp $ .

Especially, notice that the spherical harmonics $(Y_{m,j})$ belong to $C^\infty (\mathbb {S}^{n-1})$ . When we are considering the numerical algorithm in Section 3, we represent the support function in finite-dimensional function space generated by the spherical harmonics in each step N, and (2.3) will be useful. Moreover, (2.3) also suggests the relation between mixed discriminant and mixed volume, which goes back to Hilbert–Minkowski–Aleksandrov (see [1] for reference).

2.4 Sobolev space $H_k$

Let $k\ge 0$ be an integer. We denote $H_k(\mathbb S^{n-1})$ to be the Sobolev space, which is the completion of $C^\infty (\mathbb {S}^{n-1})$ with respect to the norm

$$\begin{align*}\|f\|_{H_k}=\sum_{m=0}^{k}\left(\int_{\mathbb{S}^{n-1}}\left|\nabla^{m} f\right|{}^{2} d u\right)^{1/2}, \quad f\in C^\infty(\mathbb{S}^{n-1}),\end{align*}$$

where $\nabla ^m f$ denotes the mth covariant derivative of f. For a function $g\in H_k(\mathbb {S}^{n-1})$ and $m\le k$ , $\nabla ^m g\in L^2(\mathbb {S}^{n-1})$ denotes its weak covariant derivative, which can be defined as the limit of $\nabla ^m f_k$ in $L^2$ , where $f_k\in C^\infty (\mathbb {S}^{n-1})$ ( $k\in \mathbb {N}$ ) is an arbitrary Cauchy sequence with respect to the norm $\|\cdot \|_{H_k}$ . See [18] for reference.

Let $\Delta _S: C^\infty (\mathbb {S}^{n-1})\to C^\infty (\mathbb {S}^{n-1})$ be the Laplace operator on $\mathbb {S}^{n-1}$ , which is self-adjoint. Extend it to $\Delta _S: H_{k+2}(\mathbb {S}^{n-1})\to H_k(\mathbb {S}^{n-1}).$ Then, for $f,g\in H_{k+2}(\mathbb {S}^{n-1})$ , it satisfies

$$\begin{align*}(\Delta_S f, g) = (f,\Delta_S g).\end{align*}$$

Note that $(\cdot ,\cdot )$ always denotes the scalar product in $L^2(\mathbb {S}^{n-1})$ in this paper.

2.5 Spherical harmonics

The spherical harmonics are the eigenvectors (eigenfunctions) of spherical Laplace-Beltrami operator $\Delta _S:C^\infty (\mathbb {S}^{n-1})\to C^\infty (\mathbb {S}^{n-1})$ . It is well known that these eigenvectors can be chosen to be the restriction onto $\mathbb {S}^{n-1}$ of the harmonic polynomials (with respect to an orthonormal basis $e_1,\ldots ,e_n$ of $\mathbb {R}^n$ ) on $\mathbb {R}^n\backslash \{0\}.$ The eigenvalues of $\Delta _S$ are

$$\begin{align*}\lambda_m = -m(m+n-2),\quad m=0,1,\ldots.\end{align*}$$

With regard to $\lambda _m$ , the related eigenspace $E_m$ consists of all the harmonic polynomials on $\mathbb {R}^n\backslash \{0\}$ of degree m. The dimension of $E_m$ is

$$\begin{align*}N_m =\frac{(2m+n-2)\cdot(n+m-3)!}{(n-2)!\cdot m!}.\end{align*}$$

The compact operator theory in functional analysis tells that one can choose the eigenvectors to be a Hilbert basis of $L^2(\mathbb {S}^{n-1}).$ More precisely, we can choose $(Y_{m,j})$ such that the eigenspace of $\lambda _m$ is

$$\begin{align*}E_m = \mathrm{span} \Big\{Y_{m,1},\ldots,Y_{m,N_m} \Big\}\end{align*}$$

and

$$\begin{align*}\big(Y_{m,j},Y_{m,k}\big)=\int_{\mathbb{S}^{n-1}} Y_{m,j}(u)Y_{m,k}(u) du=\delta_{j}^{k}.\end{align*}$$

For distinct $m,m'$ , we automatically have $\big (Y_{m,j},Y_{m',j'}\big ) = 0.$ Especially, when $m=1$ , the eigenfunctions $\{Y_{1,j}\}_{j=1}^n$ are chosen to be

$$\begin{align*}Y_{1,j}(u) = \frac1{\sqrt{\omega_n}} e_j\cdot u,\quad \quad j=1,\ldots,n,\end{align*}$$

so that $(Y_{1,k},Y_{1,j})=\delta _j^k.$ From this, it is easy to obtain the following formula that describes the barycenter of an integrable function g:

(2.4) $$ \begin{align} v\cdot\Big( \frac1{\omega_n} \int_{\mathbb{S}^{n-1}} u g(u)du\Big) =& \sum_{j=1}^n \Big[\int_{\mathbb{S}^{n-1}}\Big(\frac{e_j}{\sqrt{\omega_n}}\cdot u\Big) g(u) du\Big] \Big(\frac{e_j}{\sqrt{\omega_n}}\cdot v\Big)\notag\\ =& \sum_{j=1}^n (g,Y_{1,j})Y_{1,j}(v). \end{align} $$

Since $(Y_{m,j})$ is a Hilbert basis, for each $g\in L^2(\mathbb S^{n-1})$ , we have

$$ \begin{align*}g=\sum_{m=0}^\infty\sum_{j=1}^{N_m}\left(g, Y_{m,j}\right) Y_{m,j}.\end{align*} $$

3 Reconstruction from a pre-given surface area measure

For a given finite Borel measure $\mu $ that is not concentrated in any hemisphere and satisfies

$$\begin{align*}\int_{\mathbb{S}^{n-1}} vd\mu(v) = 0,\end{align*}$$

we shall give a computational procedure to reconstruct the convex body whose surface area measure is $\mu $ , and its convergence will also be proved.

For our convenience, only in this section, we relabel the spherical harmonics $\{Y_{m,j}: m=0,1,\ldots , j=1,\ldots ,N_m\}\setminus \{Y_{1,1},\ldots ,Y_{1,n}\}$ by $\{Y_m\}_{m=0}^\infty $ . That is to say, in $\{Y_i\}_{i=0}^\infty $ , there are no spherical harmonics of degree $1$ . As a result, if $h_Q = \sum _{i=0}^N x_i Y_i$ is a support function of a convex body Q. Then, by (2.4),

$$\begin{align*}\int_{\mathbb{S}^{n-1}} v h_Q(v) dv = 0.\end{align*}$$

So, one convenience is to assign a specific translation of Q, since the surface area measure $S(Q,\cdot )$ is translation invariant.

Denote

(3.1) $$ \begin{align}\mathrm{V}_{i_1,\ldots,i_n} = \frac1n\int_{\mathbb{S}^{n-1}} Y_{i_n}(v)~ \mathrm{D}(D^2 Y_{i_1},\ldots,D^2 Y_{i_{n-1}})(v)~dv,\quad i_1,\ldots,i_n=0,1,\ldots,N,\end{align} $$

and

(3.2) $$ \begin{align}\mathrm{V}_i = \int_{\mathbb{S}^{n-1}} Y_i(v) d\mu(v),\quad i=0,1,\ldots,N.\end{align} $$

Here, $\mathrm {D}$ is the mixed discriminant that can be computed by (2.2).

For a point $(x_0,\ldots ,x_N)\in \mathbb {R}^{N+1},$ let V be the volume function

$$\begin{align*}V(x_0,\ldots,x_N) = \sum_{i_1=0}^N\ldots\sum_{i_n=0}^N \mathrm{V}_{i_1,\ldots,i_n}\cdot x_{i_1}\ldots x_{i_n}.\end{align*}$$

3.1 Optimization problem ( $\overline P_N$ )

Originally, our aim in Step N is to solve the following optimization problem ( $P_N$ ):

$$\begin{align*}\max ~ V(x_0,\ldots,x_N), \end{align*}$$

$$\begin{align*}\mathrm{subject~to} \quad(x_0,\ldots,x_N)\in \overline{\mathcal{C}}_N ,\end{align*}$$

where $\mathcal {C}_N$ is a convex set (will be proved later) defined by

$$\begin{align*}\overline{\mathcal{C}}_N = \left\{ (x_0,\ldots,x_N)\in \mathbb{R}^{N+1}: { D^2\left(\sum_{m=0}^N x_m Y_m \right)(u) \geq 0,~ \forall u\in\mathbb{S}^{n-1}}, \sum_{m=0}^N \mathrm{V}_m\cdot x_m =1 \right\}.\end{align*}$$

Solve the optimization problem ( $\overline P_N$ ) and normalize the solution.

However, the problem ( $P_N$ ) cannot be solved directly by computer, since the requirement $(x_0,\ldots ,x_N)\in \overline {\mathcal {C}}_N $ has actually infinitely many inequality constraints, although it can be shown to be a convex optimization problem. Therefore, we consider the following optimization problem ( $P_{N,k}$ ), for $N,k\in \mathbb {N}$ . We will show in Theorem 3.3 that the subproblem ( $P_{N,k}$ ) is also a convex optimization problem with only finitely many constraints when k is large, and it follows from Theorems 3.3 and 3.5 that its solution will converge to the solution of the Minkowski problem as $N,k\to \infty $ .

Now, we put our computational procedure Mink-Pro and the subproblem ( $P_{N,k}$ ) as follows.

For each $k\in \mathbb {N}$ , we assign a kth partition $T_k = \{O_1^k,\ldots ,O_{2^k}^k\}$ of $\mathbb {S}^{n-1}$ such that

(3.3) $$ \begin{align} O_i^k \subset \mathbb{S}^{n-1}, \quad \mathcal{H}^{n-1}(O_i^k) = \frac{n\omega_n}{2^k}, \quad \bigcup_{i=1}^{2^k} O_i^k = \mathbb{S}^{n-1}, \quad \mathrm{and} \quad T_{k+1} \subset T_k.\end{align} $$

Extract one point from each $O_i^k$ to obtain a set $U_k\subset \mathbb {S}^{n-1}$ such that $U_k$ contains $2^k$ elements and

(3.4) $$ \begin{align}U_{k+1}\subset U_k.\end{align} $$

Define

$$\begin{align*}\mathcal{C}_{N} \kern1pt{=}\kern1pt \Big\{ (x_0,\ldots,x_N)\in \mathbb{R}^{N+1} : D^2\left(\sum_{i=0}^N x_i Y_i\right) (u) \kern1pt{\geq}\kern1pt \frac IN , ~ \forall u\kern1pt{\in}\kern1pt \mathbb{S}^{n-1}, \mathrm{~and~}~ \sum_{m=0}^N \mathrm{V}_m\cdot x_m \kern1pt{=}\kern1.5pt 1 \!\Big\},\end{align*}$$

and

(3.5) $$ \begin{align}\mathcal{C}_{N,k} \kern1pt{=}\kern1pt \Big\{ (x_0,\ldots,x_N)\in \mathbb{R}^{N+1} : D^2\left(\sum_{i=0}^N x_i Y_i\right) (u) \kern1pt{\geq}\kern1pt \frac IN , ~ \forall u\kern1pt{\in}\kern1pt U_k, \mathrm{~and~}~ \sum_{m=0}^N \mathrm{V}_m\cdot x_m \kern1pt{=}\kern1.5pt 1 \!\Big\}. \end{align} $$

Clearly, $\mathcal {C}_{N,k+1}\subset \mathcal {C}_{N,k}$ , and

$$\begin{align*}\bigcap_{k=1}^\infty\mathcal{C}_{N,k} = \mathcal{C}_{N}.\end{align*}$$

Consider the modified optimization problem ( $ P_N^*$ ):

$$\begin{align*}\max ~ V(x_0,\ldots,x_N), \end{align*}$$

$$\begin{align*}\mathrm{subject~to} \quad(x_0,\ldots,x_N)\in \mathcal{C}_N.\end{align*}$$

Denote its solution by $(x_0^*,\ldots ,x_N^*)$ , and define

(3.6) $$ \begin{align}h_{N} = \left(\sum_{i=0}^N x_i^*Y_i\right) V(x_0^*,\ldots,x_N^*)^{- 1/({n-1})}.\end{align} $$

3.2 Minkowski Procedure

• • Input: Natural numbers $n,N,k$ , the general mixed volumes $\mathrm {V}_{i_1,\ldots ,i_n}$ and $ \mathrm {V}_i$ for the spherical harmonics $Y_0,\ldots ,Y_N$ defined by (3.1) and (3.2), the finite set $U_k$ satisfying (3.4) and (3.3).

• • Task: Compute the function $h_{N,k}$ .

• • Action:

  1. 1. Solve the following convex optimization problem ( $P_{N,k}$ ):

     $$\begin{align*}\sup ~ V (x_0,\ldots,x_N), \end{align*}$$
     $$\begin{align*}\mathrm{subject~to} \quad(x_0,\ldots,x_N)\in \mathcal{C}_{N,k}.\end{align*}$$

     Denote the solution of ( $P_{N,k}$ ) by $(x_0^k,\ldots ,x_N^k).$

  2. 2. Compute

     $$\begin{align*}h_{N,k} = \left(\sum_{i=0}^N x_i^kY_i\right) V(x_0^k,\ldots,x_N^k)^{- 1/({n-1})}.\end{align*}$$

Remark 3.1 1. We will show in Theorem 3.3 that ( $P_{N,k}$ ) and ( $P_{N}$ ) are convex optimization problems, for sufficiently large k. Thus, the maximizers $(x_0^k,\ldots ,x_0^k)$ and $(x_0^*,\ldots ,x_0^*)$ (theoretically) exist and are unique.

Especially, the approach to find the solution of ( $P_{N,k}$ ) is computationally tractable, since ( $P_{N,k}$ ) has only finitely many constraints. To write the detailed algorithm for the approach, see [7, 25] for references.

2. We will prove in Theorem 3.3(6) that the convex body, determined by $h_{N,k}$ ,

$$\begin{align*}K_{N,k}= \Big\{ z : z\cdot u \leq h_{N,k}(u), \quad \forall u\in\mathbb{S}^{n-1}\Big\}\end{align*}$$

converges to $K_N$ as $k\to \infty $ . In addition, we will prove in Theorem 3.5 that $K_N\rightarrow K$ , when $N\to \infty $ , where $K_N$ is the convex body determined by $h_N$ , and $S(K,\cdot ) = \mu $ .

3. Recalling equation (2.3), $V(x_0,\ldots ,x_N)$ is actually the volume of the convex body Q whose support function is

$$\begin{align*}h_Q=\sum_{i=0}^N x_iY_i,\end{align*}$$

for $(x_0,\ldots ,x_N)\in \mathcal {C}_{N}$ . The quantity $\sum _{i=0}^N \mathrm {V}_i\cdot x_i$ is $\int h_Q d\mu .$ This explains that Action 1 coincides with the optimization problem in solving the Minkowski problem. However, they are not completely the same, and we have to prove the convergence.

4. Although our procedure can reconstruct the body from a general measure, in practice, the smooth functions in the form

$$\begin{align*}F(u) = \sum_{i=0}^{k_0} z_i Y_i(u)\end{align*}$$

are more tractable. Since the $1$ -degree spherical harmonics $Y_{1,1},\ldots ,Y_{1,n}$ are removed from $Y_i$ ’s, such a function F always satisfies $\int _{\mathbb {S}^{n-1}} uF(u) du =0$ .

Lemma 3.1 Suppose K is a convex body such that

$$\begin{align*}\int_{\mathbb{S}^{n-1}} h_K(u)d\mu(u) = 1\end{align*}$$

and

(3.7) $$ \begin{align}\int_{\mathbb{S}^{n-1}} uh_K(u)du = 0.\end{align} $$

Then, there is a constant $c_1$ that only depends on $\mu $ , such that $\max \{|x|: x\in K\} \le c_1.$

Proof Since $\mu $ is not concentrated in any hemisphere, the function

$$\begin{align*}v \mapsto\int_{S^{n-1}} (u\cdot v)_+ d\mu(u) \end{align*}$$

is strictly positive on $S^{n-1}$ , and since the function is continuous on $S^{n-1}$ , which is compact, there exists $c_0>0$ such that

$$\begin{align*}\int_{S^{n-1}} (u\cdot v)_+ d\mu(u) \geq c_0> 0\qquad\text{for all}\ v\in S^{n-1}. \end{align*}$$

Denote $R_0 = \max \{|x| : x\in \bar K_N \}.$ Then, there exists $v_0\in \mathbb {S}^{n-1}$ such that $R_0\cdot v_0\in K.$ By (3.7), we must have $o\in K.$ Thus, we obtain

$$\begin{align*}R_0\cdot c_0\le\int_{\mathbb{S}^{n-1}} R_0( v_0 \cdot u)_+ d\mu(u) \leq \int_{\mathbb{S}^{n-1}}h_{K}(u)d\mu(u)= 1,\end{align*}$$

and hence $R_0\leq 1/c_0$ . Denote $c_1=1/c_0$ , and we complete the proof.

Lemma 3.2 (1) The sets $\overline {\mathcal {C}}_N, \mathcal {C}_N, \mathcal {C}_{N,k}\subset \mathbb {R}^{N+1}$ are nonempty, closed, and convex.

(2) $\overline {\mathcal {C}}_N$ is bounded.

Proof (1) Since $Y_0$ is a constant, $(c,0,\ldots ,0)\in \mathcal {C}_N$ for some sufficiently large constant c. Since $\mathcal {C}_N \subset \overline {\mathcal {C}}_N$ and $\mathcal {C}_N\subset \mathcal {C}_{N,k}$ , they are all nonempty. Since $ D^2\Big (\sum _{i=0}^N x_iY_i\Big ) (u)$ is continuous in $(x_0,\ldots ,x_N)$ , $\overline {\mathcal {C}}_N, \mathcal {C}_N$ , and $\mathcal {C}_{N,k}$ are closed.

If, at some $u\in \mathbb {S}^{n-1}$ ,

$$\begin{align*}D^2\left(\sum_{i=0}^N x_iY_i\right) (u) \quad \mathrm{~and~} \quad D^2\left(\sum_{i=0}^N y_iY_i\right) (u)\end{align*}$$

are positive semidefinite, then

$$\begin{align*}D^2\left(\sum_{i=0}^N \big((1-\lambda)x_i+\lambda y_i\right)Y_i\Big) (u)\end{align*}$$

is also positive semidefinite at this u, $\forall \lambda \in [0,1]$ . It follows immediately that $\overline {\mathcal {C}}_N, \mathcal {C}_N$ , and $\mathcal {C}_{N,k}$ are convex.

(2) By Lemma 3.1, for any $(x_0,\ldots ,x_N)\in \mathcal {C}_N$ , we have

$$\begin{align*}|h_Q(u)| = \left|\sum_{m=0}^N x_mY_m(u)\right| \leq c_1, \quad \forall u\in\mathbb{S}^{n-1},\end{align*}$$

where $c_1$ is a constant that only depends on $\mu $ . Then,

$$ \begin{align*}|x_m|=|(h_Q, Y_m)| = \Big|\int_{\mathbb{S}^{n-1}} h_Q(u) Y_m(u) du\Big| \le c_1 \|Y_m\|_1.\end{align*} $$

Since $Y_m\in C^\infty (\mathbb {S}^{n-1})$ are fixed functions, we deduce that $\overline {\mathcal {C}}_N$ is bounded.

Theorem 3.3 (1) $V(x_0,\ldots ,x_N)$ is strictly positive on $\mathcal {C}_N$ .

(2) The function $V^{1/n}$ is strictly concave on $\mathcal {C}_{N}$ .

(3) $\mathcal {C}_{N,k}\to \mathcal {C}_N$ in the Hausdorff metric; and $\mathcal {C}_N \subset \mathrm {int}~\overline {\mathcal {C}}_N$ .

(4) $(V)^{1/n}$ is strictly concave on $\mathcal {C}_{N,k}$ , for sufficiently large k.

(5) $(x_0^k,\ldots ,x_N^k)\to (x_0^*,\ldots ,x_N^*)$ as $k\to \infty $ .

(6) The convex body $K_{N,k}$ determined by $h_{N,k}$ converges to the convex body $K_N$ determined by $h_N$ (defined in (3.6)) as $k\to \infty $ .

Proof (1) We will use the fact that a smooth support function is either the support function of a convex body or the support function of a fixed point. By our construction, $\sum _{i=0}^N x_i Y_i$ is always smooth, and $\{Y_i\}_{i=0}^N$ does not contain the spherical harmonics of degree $1$ . Thus, for $(x_0,\ldots ,x_N)\in \mathcal {C}_N$ , it cannot be the support function of a fixed point. Thus, V is always positive.

(2) Since $(x_0,\ldots ,x_N)\in \mathcal {C}_N$ ,

$$\begin{align*}\sum_{i=0}^N x_iY_i(u)\end{align*}$$

is a support function of some convex compact set, which we denote by $K(x_0,\ldots ,x_N)$ . Now,

$$ \begin{align*}V(x_0,\ldots,x_N) = V\big(K(x_0,\ldots,x_N)\big)\end{align*} $$

is its volume. Notice also that, for $(y_0,\ldots ,y_N)\in \mathcal {C}_N$ and $\lambda \in [0,1],$

$$\begin{align*}(1-\lambda)K(x_0,\ldots,x_N) + \lambda K(y_0,\ldots,y_N) = K\big((1-\lambda) x_0+ \lambda y_0,\ldots,(1-\lambda) x_N+ \lambda y_N\big).\end{align*}$$

Thus, statement (2) follows immediately from the Brunn–Minkowski inequality, statement (1), and the fact that

$$\begin{align*}\int_{\mathbb{S}^{n-1}} u h_{K(x_0,\ldots,x_N)}du = \int_{\mathbb{S}^{n-1}} uh_{K(y_0,\ldots,y_N)}(u)du =0.\end{align*}$$

(3) Since $\mathcal {C}_{N,k}$ is convex and closed for each $k\in \mathbb {N}$ ,

$$\begin{align*}\bigcap_{k=1}^\infty\mathcal{C}_{N,k} = \mathcal{C}_{N},\end{align*}$$

and $\mathcal {C}_N$ is compact, statement (3) follows immediately.

(4) Since $\mathcal {C}_{N,k} \to \mathcal {C}_N\subset \mathrm {int} \overline {\mathcal {C}}_N$ , and $ \overline {\mathcal {C}}_N$ is compact, V is still positive on $\mathcal {C}_{N,k}$ when k is sufficiently large. Moreover, $\mathcal {C}_{N,k}$ will also be compact. Recall that

$$\begin{align*}V(x_0,\ldots,x_N) = \sum_{i_1=0}^N\ldots\sum_{i_n=0}^N \mathrm{V}_{i_1,\ldots,i_n}\cdot x_{i_1}\ldots x_{i_n}\end{align*}$$

is a polynomial of degree n, where $\mathrm {V}_{i_1,\ldots ,i_n}$ are fixed constants. Thus, $V(\cdot ) C^\infty (\mathbb {R}^{N+1})$ . By the construction of $\mathcal {C}_{N,k}$ , when k is sufficiently large, there is a constant $\theta>0,$ such that the Hessian

$$ \begin{align*}D^2(V^{1/n})(x_0,\ldots,x_n)\ge \theta I_{N+1}\end{align*} $$

everywhere on $\mathcal {C}_{N,k}$ .

(5) As a consequence of (3), for sufficiently large k, $\mathcal {C}_{N,k}$ is also compact. Thus, any subsequence of $(x_0^k,\ldots ,x_N^k)$ has a subsubsequence, which is also denoted by $(x_0^k,\ldots ,x_N^k)$ , converging to a point $(\bar x_0,\ldots ,\bar x_N) \in \bigcap _k \mathcal {C}_{N,k} =\mathcal {C}_N$ . It suffices to show that $(\bar x_0,\ldots ,\bar x_N)=(x_0^*,\ldots ,x_N^*)$ .

Since $(x_0^*,\ldots ,x_N^*)\in \mathcal {C}_{N,k}$ , for each $k\in \mathbb {N}$ , and $(x_0^k,\ldots ,x_N^k)$ is the maximizer of ( $P_{N,k}$ ) for sufficiently large k, we have

(3.8) $$ \begin{align} V (x_0^k,\ldots,x_N^k) \geq V (x_0^*,\ldots,x_N^*). \end{align} $$

Taking limits in (3.8), we obtain

$$\begin{align*}V (\bar x_0,\ldots,\bar x_N) \geq V (x_0^*,\ldots,x_N^*).\end{align*}$$

Since $(\bar x_0,\ldots ,\bar x_N)\in \mathcal {C}_N$ , and $(x_0^*,\ldots ,x_N^*)$ is the unique maximizer of ( $P_N^*$ ), we must have $(\bar x_0,\ldots ,\bar x_N)=(x_0^*,\ldots ,x_N^*)$ . This completes the proof.

(6) By (5), we have

$$\begin{align*}\sum_{i=0}^N x_i^k Y_i \to \sum_{i=0}^N x_i^* Y_i\end{align*}$$

uniformly on $\mathbb {S}^{n-1}$ . Recall the fact collected in Section 2.1: $f_i\to f$ uniformly and that $W[f]$ has interior imply that $W[f_i]\to W[f]$ in the Hausdorff metric. Now, (6) follows immediately from this property, the continuity of volume, and the definitions of $h_N$ and $h_{N,k}$ .

Lemma 3.4 [29, Theorems 4 and 5]

Suppose $F\in C^\infty (\mathbb {S}^{n-1})$ satisfying $\int F(u)udu =0.$ Denote

$$\begin{align*}F_N(u)= \sum_{m=0}^N (F,Y_{m})(u).\end{align*}$$

Then:

1. (1) $F_N$ converges to F uniformly on $\mathbb {S}^{n-1}$ ;

2. (2) $D^2F_N$ converges to $D^2F$ uniformly on $\mathbb {S}^{n-1}$ .

Remark 3.2 This Lemma follows immediately from the estimates in [29, Theorems 4 and 5], which suggest the relationship between the max norm and the $L^2$ -norm of Fourier expansion into spherical harmonics. The assumption $\int F(u)udu =0$ is because our $\{Y_i\}_{i=0}^\infty $ does not contain the spherical harmonics $Y_{1,1},\ldots ,Y_{1,n}$ . Except for this, the only difference is that the $D^\alpha $ in [29, Theorem 4(b)] is slightly different from ours.

Let $f\in C^\infty (\mathbb {S}^{n-1})$ . Denote $\bar D^2 f(x/|x|)$ to be the Euclidean Hessian, and $D^2 f$ to be our notation in Section 2.1. Then, restricting to $x^\perp $ ,

$$\begin{align*}\bar D^2 f(x/|x|) + f(x/|x|)I = D^2 f(x/|x|),\end{align*}$$

and $\bar D^2 f(x/|x|) =0$ on the line $\mathbb {R} x$ .

Theorem 3.5 Denote the solution of ( $P_N$ ) by $(x_0^*,\ldots ,x_N^*),$ and

$$\begin{align*}h_N = \left(\sum_{m=0}^N x_mY_m\right) V(x_0^*,x_1^*,\ldots,x_N^*)^{- 1/({n-1})}.\end{align*}$$

Let $K_N$ be the convex body determined by $h_N$ ,

$$\begin{align*}K_N = \{x: x\cdot u \le h_N(u),\ \forall u\in\mathbb{S}^{n-1}\}.\end{align*}$$

Then, $K_N$ converges to K in the Hausdorff metric, as $N\to \infty $ . Here, K is the unique (theoretical) solution of the Minkowski problem

$$\begin{align*}S(K,\cdot) = \mu(\cdot)\end{align*}$$

such that

$$\begin{align*}\int_{\mathbb{S}^{n-1}} uh_K(u)du=0.\end{align*}$$

Proof Denote

$$\begin{align*}h_{\bar K_N} = \sum_{m=0}^N x_m^*Y_m,\end{align*}$$

and denote $\bar K=\lambda K$ ( $\lambda>0$ ) to be the dilation of K such that $\int _{\mathbb {S}^{n-1}} h_{\bar K} d\mu = 1.$ To prove this theorem, it suffices to prove that $\bar K_N\to \bar K$ .

Step 1. Since $\{Y_m\}_{m=0}^\infty $ does not contain the $\{Y_{1,j}\}$ ’s, by equation (2.4), we always have

(3.9) $$ \begin{align}\int_{\mathbb{S}^{n-1}}uh_{\bar K_N}(u)du=0.\end{align} $$

Step 2. It follows immediately from Lemma 3.1 that the sequence $\{\bar K_N\}_{N=1}^\infty $ is bounded.

Step 3. By Step 2 and Blaschke’s selection theorem, any subsequence of $\{\bar K_N\}_{N=1}^\infty $ has a subsubsequence $\bar K_{N_i}$ converging to a convex body $K^*$ in the Hausdorff metric. In this step, we will show that $K^*$ satisfies

(3.10) $$ \begin{align}\qquad V(K^*) = \sup \Big\{V(Q): \int_{\mathbb{S}^{n-1}} h_Q(u)d\mu(u) =1,\ Q\ \mathrm{is\ a\ convex\ body}\Big\}, \end{align} $$

and

(3.11) $$ \begin{align}\int_{\mathbb{S}^{n-1}} h_{K^*}(u)d\mu(u)=1.\end{align} $$

The second equation follows immediately from the convergence. Let us consider (4.10).

• • Step 3.1. Suppose Q is a smooth convex body that has positive Gauss curvature, and $\int _{\mathbb {S}^{n-1}}h_Qd\mu =1$ . We will show that $V(K^*)\ge V(Q).$ By the assumption, $h_Q\in C^\infty (\mathbb {S}^{n-1})$ , and

  $$\begin{align*}\det (D^2h_Q)(u)> 0 , \quad \forall u\in\mathbb{S}^{n-1}.\end{align*}$$
  Denote
  $$\begin{align*}h_{Q_N} = \sum_{m=0}^N (h_Q,Y_{m})Y_m.\end{align*}$$
  Then, by Lemma 3.4(1), $h_{Q_N}$ converges to $h_Q$ uniformly. In addition, by Lemma 3.4(2), for sufficiently large N, we still have
  $$\begin{align*}\det (D^2h_{Q_N})(u)> 0, \quad \forall u\in\mathbb{S}^{n-1},\end{align*}$$
  and hence $h_{Q_N}$ is also support function. Therefore, if we denote
  $$\begin{align*}\bar Q_N =\frac1{\int_{\mathbb{S}^{n-1}}h_{Q_N}d\mu} Q_N, \end{align*}$$
  by the construction of $\bar K_N$ , we have $V(\bar K_N)\ge V(\bar Q_N)$ . Since $\bar K_N\to K^*$ , $\bar Q_N\to Q$ , we get
  $$\begin{align*}V(K^*)\ge V(Q).\end{align*}$$

• • Step 3.2. For any convex body L, we can choose a sequence of smooth convex bodies $Q_m$ that have positive curvatures, and such that $Q_m\to L$ . By the approximation argument, we complete the proof of (4.10).

By the solution of the Minkowski problem, the convex body satisfying (4.10) and (3.11) is unique up to translations. Recalling the fact that

$$\begin{align*}\int_{\mathbb{S}^{n-1}}uh_{\bar K}(u)du = \int_{\mathbb{S}^{n-1}}uh_{ K^*}(u)du =0,\end{align*}$$

we have $\bar K= K^*$ . Since we have proved that any subsequence of $\{\bar K_N\}$ has a convergent subsequence converging to $\bar K$ , we have actually completed the proof of this theorem.

4 Reconstruction problems (II) for the lightness function

In this section, we consider the general lightness function $Q_K,$ which was defined by

$$\begin{align*}Q_K(v,w) = \int_{\mathbb{S}^{n-1}} (v\cdot u)_- f_w(u) dS(K,u),\end{align*}$$

where $f_w$ is positive and continuous in the open set $w^+\cap \mathbb {S}^{n-1}$ and vanishes in its complement. Before this, we need to develop some applications about the Funk–Hecke theorem.

4.1 Further applications of the Funk–Hecke theorem

The following is known as the Funk–Hecke theorem.

Lemma 4.1 [28]

Let $f \in L_2[-1,1]$ be a function satisfying

$$ \begin{align*} \int^{1}_{-1}|f(t)|(1-t^2)^{\frac{n-3}{2}}<\infty. \end{align*} $$

If $Y_m \in \mathscr {F}^m$ is a spherical harmonic of degree m, then

$$\begin{align*} \int_{\mathbb{S}^{n-1}}f(u\cdot v) Y_m(u)du=\lambda_{m}(f)Y_m(v), \end{align*}$$

for $v\in \mathbb {S}^{n-1}$ , where

$$\begin{align*} \lambda_{m}(f)=(n-1)\omega_{n-1}[C_m^\nu(1)]^{-1}\int_{-1}^{1}f(t)C_m^\nu(t)(1-t^2)^{\frac{n-3}{2}}dt, \end{align*}$$

where

$$ \begin{align*}C_{m}^{\nu}(t)=a_{m}^{\nu}\left(1-t^{2}\right)^{-\nu+1 / 2}\left(\frac{d}{d t}\right)^{m}\left(1-t^{2}\right)^{m+\nu-1 / 2},\end{align*} $$

$$\begin{align*} a_{m}^{\nu}=C_{m}^{\nu}(1)\left(-\frac{1}{2}\right)^{m} \frac{\Gamma\left(\frac{n-1}{2}\right)}{\Gamma\left(m+\frac{n-1}{2}\right)},\quad \mathrm{and}\quad C_{m}^{\nu}(1)=\frac{\Gamma(m+n-2)}{\Gamma(n-2) \Gamma(m+1)}.\end{align*}$$

Lemma 4.2 [28]

For the function $f(t)=|t|$ , the eigenvalue $\lambda _m(f)$ in the Funk–Hecke theorem is

$$ \begin{align*} \lambda_{m}(|t|)=\frac{(-1)^{\frac{m-2}{2}} \pi^{\frac{n-1}{2}}\Gamma{(m-1)}}{2^{m-2}\Gamma\left(\frac{m }{2}\right) \Gamma\left(\frac{m+n+1}{2}\right)}, \end{align*} $$

for even m, and $\lambda _{m}(|t|)=0$ for odd m. Furthermore, $|\lambda _m(|t|)^{-1}|=O\left (m^{(n+2)/2}\right )$ for even $m\rightarrow \infty .$

Since the lightness functions are defined in the form of partial cosine transform

$$\begin{align*}\mathcal{C}_- g(v) = \int_{\mathbb{S}^{n-1}}(v\cdot u)_-g(u)du,\end{align*}$$

we require the following Lemma to compute its eigenvalues.

Lemma 4.3 Let $f(t)=t_-$ , and denote $\beta _m $ to be the $\lambda _m(f)$ in the Funk–Hecke theorem. Then,

$$ \begin{align*} \beta_{m}=\frac{(-1)^{\frac{m-2}{2}} \pi^{\frac{n-1}{2}}\Gamma{(m-1)}}{2^{m-1}\Gamma\left(\frac{m }{2}\right) \Gamma\left(\frac{m+n+1}{2}\right)}, \end{align*} $$

for even m, $\beta _{m}=0$ if m is odd and $m\geq 3,$ and

$$ \begin{align*}\beta_1 =-\frac{\omega_n}2 \neq 0.\end{align*} $$

Applying Lemma 4.1, one can directly compute the coefficients $\beta _m$ , but it will be complicated when $m\ge 3$ is odd. However, we would like to provide the following simpler method, to understand the geometric fact that the partial cosine transform $\mathcal {C}_- g$ also defines a support function of a zonoid (which may not be symmetric with respect to the origin).

Proof Since $t_-= 1/2(|t| - t)$ , we have

(4.1) $$ \begin{align}\int_{\mathbb{S}^{n-1}} (v\cdot u)_- g(u) du =& \frac12\int_{\mathbb{S}^{n-1}} |v\cdot u| g(u) du - \frac12\int_{\mathbb{S}^{n-1}} (v\cdot u)g(u)du \notag\\ =& \frac12\int_{\mathbb{S}^{n-1}} |v\cdot u| g(u) du + v\cdot\Big( -\frac12\int_{\mathbb{S}^{n-1}} u g(u)du\Big). \end{align} $$

Denote $a_{m,j} = (g,Y_{m,j})$ . By the above equation and Lemmas 4.1 and 4.2, we have

(4.2) $$ \begin{align}\int_{\mathbb{S}^{n-1}} (v\cdot u)_- g(u) du = \frac12\sum_{m~\mathrm{ even}}\sum_{j=1}^{N_m}\lambda_m(|t|)a_{m,j}Y_{m,j}(v) - \frac12\int_{\mathbb{S}^{n-1}} (v\cdot u)g(u)du.\end{align} $$

Using Lemma 4.1 again, for the function $(v\cdot u)_-,$ we know that

(4.3) $$ \begin{align}\int_{\mathbb{S}^{n-1}} (v\cdot u)_- g(u) du = \sum_{m=0}^\infty\sum_{j=1}^{N_m}\beta_m a_{m,j}Y_{m,j}(v).\end{align} $$

By (2.4), we have

$$ \begin{align*} -\frac12\int_{\mathbb{S}^{n-1}} (v\cdot u) g(u)du =& -\frac{\omega_n}2\cdot\sum_{j=1}^n a_{1,j}Y_{1,j}(v).\end{align*} $$

By this and Lemma 4.2, comparing (4.2) with (4.3), we get the desired result.

Lemma 4.4 (Theorem 3.5.4 [28])

If $\rho $ is an even signed measure on $\mathbb {S}^{n-1}$ with

$$ \begin{align*}\int_{\mathbb{S}^{n-1}}|(u\cdot v)| \mathrm{d} \rho(u)=0 \quad \forall~ v \in \mathbb{S}^{n-1}, \end{align*} $$

then $\rho =0$ .

If G is an even real function on $\mathbb {S}^{n-1}$ of differentiability class $C^k$ , where $k\geq n+2$ is even, then there exist an even continuous function g on $\mathbb {S}^{n-1}$ such that

(4.4) $$ \begin{align} G(v)=\int_{\mathbb{S}^{n-1}}|(u\cdot v)|g(u)du, \end{align} $$

for $v\in \mathbb {S}^{n-1}$ .

The above Lemma and its proof can be found in [28]. Since we want to work with the nonsmooth convex bodies, we slightly improve this result to the functions in Sobolev spaces. Precisely, we consider the functions in ${H}_{k}(\mathbb {S}^{n-1})$ . Although the approach is standard, we did not find the result as follows. So, for completeness, we give a proof here.

Lemma 4.5 Let $k\ge n+1$ be an even integer. If $G \in H_k(\mathbb S^{n-1})$ is an even function, then there exists an even function $g\in L^2(\mathbb {S}^{n-1})$ such that

(4.5) $$ \begin{align} G(v)=\int_{\mathbb{S}^{n-1}}|(u\cdot v)| g(u)du, \end{align} $$

for all $v\in \mathbb {S}^{n-1}$ . Especially, g can be formulated by the convergent function series

(4.6) $$ \begin{align} \sum_{m~\mathrm{even}} \sum_{j=1}^{N_m} \lambda_m^{-1}(|t|)(G,Y_{m,j}) Y_{m,j}, \end{align} $$

where $\lambda _m^{-1}(|t|)$ is provided by the Funk–Hecke theorem.

Proof Denote $k=2l$ . Since $G\in H_k(\mathbb {S}^{n-1}),$ we have

$$ \begin{align*}G(v)= \sum_{m=0}^{\infty}\sum_{j=1}^{N_m}(G,Y_{m,j})Y_{m,j}(v).\end{align*} $$

We aim to prove that the series

(4.7) $$ \begin{align} \sum_{ ~m~\mathrm{even}} \lambda_m(|t|)^{-1}\left(\sum_{j=1}^{N_m}(G,Y_{m,j})Y_{m,j}\right) \end{align} $$

converges in $ L^2(\mathbb {S}^{n-1})$ .

Recall that

$$ \begin{align*}\Delta_{S} Y_{m,j} = -m(m+n-2)Y_{m,j}.\end{align*} $$

Since $\Delta _{S}$ is self-adjoint, we have

$$ \begin{align*} \left(G, Y_{m,j}\right)=\left[\frac{-1}{m(m+n-2)}\right]^l \left(\Delta_{S}^{l } G, Y_{m,j}\right). \end{align*} $$

By the Cauchy–Schwarz inequality, we have

$$\begin{align*}|(G, Y_{m,j})| \leq \left[\frac{1}{m(m+n-2)}\right]^l \cdot \|G\|_{H_k}.\end{align*}$$

As a result, the right-hand side of (4.7) satisfies

$$\begin{align*}\Big\|\lambda_m(|t|)^{-1}\left(\sum_{j=1}^{N_m}(G,Y_{m,j})Y_{m,j}\right) \Big\|_2^2\leq \|G\|_{H_k}^2 \left[\frac{1}{m(m+n-2)}\right]^{2l}\cdot \lambda_m(|t|)^{-2}\cdot {N_m}.\end{align*}$$

Since $G\in H_k$ , it suffices to prove that

(4.8) $$ \begin{align} {\left[m(m+n-2)\right]}^{-2l}\cdot \lambda_m(|t|)^{-2}\cdot {N_m} =\mathrm{O}\left(m^{\alpha}\right),\quad \mathrm{~for~some~} \alpha<-1. \end{align} $$

Here,

$$\begin{align*}\lambda_m(|t|)^{-1} = \mathrm{O}\left(m^{\frac{n+2}{2}}\right), \quad \mathrm{~and~}\quad N_m= \frac{(2m+n-2)\cdot(n+m-3)!}{(n-2)!\cdot m!} = \mathrm{O}\left(m^{\frac{n-2}{2}}\right).\end{align*}$$

If $l>n/2+1/4$ , the inequality (4.8) holds for

$$ \begin{align*}\alpha = {-4l+2n} <-1.\end{align*} $$

Therefore, when $k=2l\ge n+1$ , we have the desired result.

Lemmas 4.3 and 4.5 immediately lead to the following.

Corollary 4.6 Let $k\ge n+1$ be an even integer, and let $G\in H_k(\mathbb {S}^{n-1})$ . Suppose $g\in L^2(\mathbb {S}^{n-1})$ satisfies

$$\begin{align*}\int_{\mathbb{S}^{n-1}}(v\cdot u)_- g(u) du = G(v).\end{align*}$$

Then,

$$\begin{align*}g(u) + g(-u) = 2\sum_{m~\mathrm{even}} \sum_{j=1}^{N_m} \beta_m^{-1}(G,Y_{m,j}) Y_{m,j}\end{align*}$$

and

$$\begin{align*}\int_{\mathbb{S}^{n-1}}ug(u)du = -\frac2{\omega_n}\int_{\mathbb{S}^{n-1}}uG(u)du.\end{align*}$$

Moreover, $(G,Y_{m,j}) = 0$ , for each odd integer $m\geq 3$ .

Proof Denote $\tilde g(u) = g(u) + g(-u), a_{m,j} = (g,Y_{m,j}),$ and $b_{m,j} = (\tilde g,Y_{m,j}).$ When m is even, $Y_{m,j}$ is even, and thus $b_{m,j} = 2a_{m,j}$ . On one hand, by Lemmas 4.1 and 4.2, we have

$$\begin{align*}G(v)=\int_{\mathbb{S}^{n-1}}(v\cdot u)_- g(u) du = \sum_{m~\mathrm{even}}\sum_{j=1}^{N_m} \beta_m a_{m,j}Y_{m,j}(v) -\frac{\omega_n}2\sum_{j=1}^n a_{1,j}Y_{1,j}(v). \end{align*}$$

On the other hand,

$$\begin{align*}G(v)=\sum_{m=0}^\infty\sum_{j=1}^{N_m} (G,Y_{m,j})Y_{m,j}(v). \end{align*}$$

Comparing the coefficients, we have the following. For each odd integer $m\geq 3$ , we have $(G,Y_{m,j}) = 0$ . For $m=1$ , we have $(G,Y_{1,j}) =- ({\omega _n}/2) a_{1,j}.$ Applying (2.4) for g and G, respectively, we get

$$\begin{align*}-\frac{\omega_n}2 \int_{\mathbb{S}^{n-1}} u g(u)du = \int_{\mathbb{S}^{n-1}} u G(u)du. \end{align*}$$

For even m, we have

$$\begin{align*}(G,Y_{m,j}) = \frac{\beta_m }2b_{m,j}.\end{align*}$$

Since $\tilde g$ is an even function, we get

$$\begin{align*}\tilde g(u) = g(u) + g(-u) = 2\sum_{m~\mathrm{even}} \sum_{j=1}^{N_m} \beta_m^{-1}(G,Y_{m,j}) Y_{m,j}.\end{align*}$$

The convergence of the above function series follows from Lemma 4.5 and the fact that $2\beta _m = \lambda _m(|t|)$ .

4.2 The necessary and sufficient conditions of being a lightness function

Given a binary function $G: \mathbb {S}^{n-1}\times \mathbb {S}^{n-1} \rightarrow \mathbb {R}$ so that $G(\cdot ,w)$ is Sobolev for a fixed $w\in \mathbb {S}^{n-1}$ , we shall use the notation

$$\begin{align*}\left(G(\cdot,w),f\right) = \int_{\mathbb{S}^{n-1}} G(v,w)f(v) dv.\end{align*}$$

Lemma 4.7 Let K be a convex body, and let $k\ge n+1$ be an even integer. If, for each $i\in \{1,\ldots ,n\}$ , its lightness functions $Q_K(\cdot ,e_i), Q_K(\cdot ,-e_i)\in H_k(\mathbb {S}^{n-1})$ , then the surface area measure $S(K,\cdot )$ has a density in $L^2(\mathbb {S}^{n-1})$ .

Proof Denote

$$\begin{align*}G_{K,i}(v) = Q_K(v,e_i)+Q_K(-v,e_i) + Q_K(v,-e_i)+Q_K(-v,-e_i),\end{align*}$$

which is an even function in $H_k(\mathbb {S}^{n-1})$ . On one hand,

$$\begin{align*}G_{K,i}(v) =\int_{\mathbb{S}^{n-1}}|u\cdot v| \left(f_{e_i}(u) + f_{-e_i}(u)\right) dS(K,u).\end{align*}$$

On the other hand, by Lemma 4.5, there is an even function $F_{K,i}\in L^2(\mathbb {S}^{n-1})$ such that

$$\begin{align*}G_{K,i}(v) = \int_{\mathbb{S}^{n-1}} |u\cdot v| F_{K,i}(u) du. \end{align*}$$

Since the cosine transform is injective (Lemma 4.4), we have

$$\begin{align*}\left(f_{e_i}(u) + f_{-e_i}(u)\right) dS(K,u) = F_{K,i}(u) du. \end{align*}$$

Note that, for each $u\in \mathbb {S}^{n-1},$ there exists an $i\in \{1,\ldots ,n\}$ such that $|u_i|\ge 1/\sqrt n.$ Since $|\tilde f_{e_i}|$ is positive and continuous in $\{u: |u_i|\geq 1/\sqrt n\}$ , it has a strict positive lower bound. Thus,

$$\begin{align*}dS(K,u) = \frac{F_{K,i}(u)}{f_{e_i}(u) + f_{-e_i}(u)} du\end{align*}$$

has a density in $L^2(\mathbb {S}^{n-1})$ , where the function $F_{K,i}/(f_{e_i}+ f_{-e_i})$ must be independent of i.

Theorem 4.8 Let K be a convex body, and let $k\ge n+1$ be an even integer. Suppose $Q_K(\cdot ,w ) \in H_k(\mathbb {S}^{n-1})$ , for any $w\in \mathbb {S}^{n-1}$ . Then:

• (1) The curvature function $F_K$ satisfies

  $$\begin{align*}\frac12 f_w(u)F_K(u) = 1_{w^+}(u)\cdot\sum_{m \mathrm{~even}}\sum_{j=1}^{N_m} \beta_m^{-1}\big(Q_K(\cdot,w),Y_{m,j}\big) Y_{m,j}(u).\end{align*}$$

• (2) For each odd integer $m\geq 3$ , and for each fixed $w\in \mathbb {S}^{n-1}$ , the function $Q_K(\cdot ,w ): \mathbb {S}^{n-1}\rightarrow \mathbb {R}$ satisfies

  $$\begin{align*}\big(Q_K(\cdot,w ),Y_{m,j}\big)=0, \quad \forall j=1,\ldots,N_{m}. \end{align*}$$

• (3) Moreover, $F_K$ satisfies

  $$\begin{align*}\int_{\mathbb{S}^{n-1}}vQ_K(v,w) dv = -\frac{\omega_n}2 \int_{\mathbb{S}^{n-1}} u f_w F_K(u)du , \quad \forall i=1,\ldots,n.\end{align*}$$

Proof By Lemma 4.7, $S(K,\cdot )$ has an $L^2(\mathbb {S}^{n-1})$ density $F_K:\mathbb {S}^{n-1}\rightarrow [0,\infty )$ . Substitute the function $f_w F_K$ into Corollary 4.6, and observe that

$$\begin{align*}Q_K(v,w) = \int_{\mathbb{S}^{n-1}} (v\cdot u)_- f_w(u) F_K(u) du.\end{align*}$$

Then, we immediately obtain (2) and (3), and

$$\begin{align*}\frac12\left(f_w(u) F_K(u) + f_w(-u) F_K(-u)\right) = \sum_{m \mathrm{~even}}\sum_{j=1}^{N_m} \beta_m^{-1}\big(Q_K(\cdot,w),Y_{m,j}\big) Y_{m,j}(u).\end{align*}$$

Since $f_w(u)$ is concentrated on $w^+, f_w(-u) F_K(-u)$ is just a symmetric copy of $f_w(u) F_K(u)$ in $w^-$ . Thus,

$$\begin{align*}\frac12 f_w(u) F_K(u) = 1_{w^+}(u)\cdot\sum_{m \mathrm{~even}}\sum_{j=1}^{N_m} \beta_m^{-1}\big(Q_K(\cdot,w),Y_{m,j}\big) Y_{m,j}(u).\end{align*}$$

This explains (1).

Theorem 4.9 Let $G: \mathbb {S}^{n-1}\times \mathbb {S}^{n-1} \rightarrow \mathbb {R}$ be a binary function, and let $k\ge n+1$ be an even integer. Suppose $G(\cdot ,w)\in H_k(\mathbb {S}^{n-1})$ , for any fixed $w\in \mathbb {S}^{n-1}$ . In addition, suppose G satisfies the following conditions:

• (1) The following function

  (4.9) $$ \begin{align} F(u) = \begin{cases} \sum\limits_{m~\mathrm{even}} \sum\limits_{k=1}^{N_m} \beta_m^{-1}\big(G(\cdot,w),Y_{m,k}\big) \frac{Y_{m,k}(u)}{f_w(u)} , \ \ & u\cdot w>0 \\ \phantom{\cdot} &\phantom{\cdot} \\ \sum\limits_{m~\mathrm{even}} \sum\limits_{k=1}^{N_m} \beta_m^{-1}\big(G(\cdot,-w),Y_{m,k}\big) \frac{Y_{m,k}(u)}{f_{-w}(u)} , \ \ & u\cdot w<0 \end{cases} \end{align} $$
  is nonnegative and independent of $w\in \mathbb {S}^{n-1}$ .

• (2) For each odd integer $m\geq 3$ , and for each fixed $w\in \mathbb {S}^{n-1}$ , the function $G(\cdot ,w): \mathbb {S}^{n-1}\rightarrow \mathbb {R}$ satisfies

  $$\begin{align*}\big(G(\cdot,w),Y_{m,j}\big)=0, \quad \forall k=1,\ldots,N_{m}. \end{align*}$$

• (3) The function F in (1) satisfies

  $$\begin{align*}\int_{\mathbb{S}^{n-1}} uF(u) du =0.\end{align*}$$

• (4) G and the function F in (1) satisfy

  $$\begin{align*}\frac 2{\omega_n}\int_{\mathbb{S}^{n-1}} vG(v,w)dv = \int_{\mathbb{S}^{n-1}} uf_w(u)F(u)du.\end{align*}$$

Then, there is a convex body K such that

$$\begin{align*}G = Q_K,\end{align*}$$

and K is unique up to translations.

Condition (1) means that for any $w_1,w_2\in \mathbb {S}^{n-1},$ the related two function series are equivalent almost everywhere.

Proof By Lemma 4.5, for each fixed $w\in \mathbb {S}^{n-1}$ , the function series

(4.10) $$ \begin{align}\sum_{m \mathrm{~even}} \sum_{j=1}^{N_m} \beta_m^{-1}\big(G(\cdot,w),Y_{m,j}\big) Y_{m,j}(u) \in L^2(\mathbb{S}^{n-1}).\end{align} $$

Let $e_1,\ldots ,e_n$ be an orthonormal basis of $\mathbb {R}^n$ , and denote

$$\begin{align*}\Omega_i = \Big\{ u\in\mathbb{S}^{n-1}: |u\cdot e_i| \geq 1/\sqrt n \Big\}. \end{align*}$$

By (4.9) and (4.10), we infer that $F \in L^2(\Omega _i)$ , $i=1,\ldots ,n.$ Since $\{\Omega _i\}_{i=1}^n$ covers $\mathbb {S}^{n-1}$ , we see that $F \in L^2(\mathbb {S}^{n-1}).$ Since $\mathbb {S}^{n-1}$ is compact, F is also in $L^1(\mathbb {S}^{n-1})$ . By conditions (1) and (3), F is independent of w and is of centroid $0$ . Then, there is a unique (up to translations) convex body K that solves the classical Minkowski problem

$$\begin{align*}dS(K,u) = F(u)du.\end{align*}$$

Notice that

$$\begin{align*}\frac12\left(f_w(u) F(u) + f_w(-u) F(-u)\right) = \sum_{m \mathrm{~even}}\sum_{j=1}^{N_m} \beta_m^{-1}\big(Q_K(\cdot,w),Y_{m,j}\big) Y_{m,j}(u).\end{align*}$$

For this K, by Corollary 4.6, Lemma 4.3, condition (4), equation (2.4), and finally by condition (2), we have

$$ \begin{align*} Q_K(v,w) &= \int_{\mathbb{S}^{n-1}} (v\cdot u)_- f_w(u) F(u) du\\ & = \frac12\int_{\mathbb{S}^{n-1}} (v\cdot u)_- \frac12\left(f_w(u) F(u) + f_w(-u) F(-u)\right) du \\&\quad- \frac12 \int_{\mathbb{S}^{n-1}} (u\cdot v) f_w(u) F(u) du\\ & = \int_{\mathbb{S}^{n-1}} (v\cdot u)_- \sum_{m~\mathrm{even}} \sum_{j=1}^{N_m} \beta_m^{-1} \big(G(\cdot,w),Y_{m,j}\big) Y_{m,j}(u) \\&\quad + \frac1{ {\omega_n}}\int_{\mathbb{S}^{n-1}}(u\cdot v) G(u,w)du\\ &= \sum_{m~\mathrm{even}} \sum_{j=1}^{N_m} \big(G(\cdot,w),Y_{m,j}\big) Y_{m,j}(v) + \frac1{ {\omega_n}}\int_{\mathbb{S}^{n-1}}(u\cdot v) G(u,w)du\\ &= \sum_{m~\mathrm{even}} \sum_{j=1}^{N_m} \big(G(\cdot,w),Y_{m,j}\big) Y_{m,j}(v) + \sum_{j=1}^{N_1} \big(G(\cdot,w),Y_{1,j}\big) Y_{1,j}(v)\\ &= \sum_{m=0}^\infty \sum_{j=1}^{N_m} \big(G(\cdot,w),Y_{m,j}\big) Y_{m,j}(v)\\ &= G(v,w). \end{align*} $$

That the convex body K is unique follows from Theorem 4.8 and the uniqueness of the Minkowski problem.

5 Reconstruction from the pre-given lightness function

In this section, for a pre-given lightness function $Q_K(v,w)$ together with a function $f_w$ as in (1.2), we give the following numerical procedure to reconstruct K.

Light-Pro

• • Input: Natural numbers $n,M,N,k$ , the spherical harmonics $Y_{m,j}$ , where $m=1,\ldots ,M$ , and $j=1,\ldots ,N_m$ .

• • Task: Compute the function $h_{M,N,k}$ .

• • Action:

  1. 1. Fix a $w\in \mathbb {S}^{n-1}$ . Compute

     $$\begin{align*}A_{m,j} = \beta_m^{-1} (Q(\cdot,w), Y_{m,j})\end{align*}$$
     and
     $$\begin{align*}B_{m,j} = \beta_m^{-1} (Q(\cdot,-w), Y_{m,j}).\end{align*}$$

  2. 2. Compute

     (5.1) $$ \begin{align} F_M(u) = \begin{cases} \sum\limits_{\substack{m=0\\m~\mathrm{even}}}^M \sum\limits_{k=1}^{N_m} A_{m,j} \frac{Y_{m,k}(u)}{f_w(u)} , \ \ & u\cdot w>0, \\ \phantom{\cdot} &\phantom{\cdot} \\ \sum\limits_{\substack{m=0\\m~\mathrm{even}}}^M \sum\limits_{k=1}^{N_m} B_{m,j}\frac{Y_{m,k}(u)}{f_{-w}(u)} , \ \ & u\cdot w<0. \end{cases} \end{align} $$

  3. 3. Compute

     (5.2) $$ \begin{align}\tilde F_M(u) = F_M(u) - \frac1{\omega_n}\int_{\mathbb{S}^{n-1}} (u\cdot v) F_M(v) dv.\end{align} $$

  4. 4. If $\tilde F_M\ge 0,$ then apply the Mink-Pro to find the solution of the ( $P_{N,k}$ ) problem for

     $$\begin{align*}d\mu =\tilde F_M du,\end{align*}$$
     and denote the resulting function by $h_{M,N,k}$ .

Remark 5.1 1. On one hand, it follows from a direct computation that the $\tilde F_M$ defined in (5.2) satisfies

$$\begin{align*}\int_{\mathbb{S}^{n-1}} u\tilde F_M(u) du=0.\end{align*}$$

On the other hand, since $f_w$ is positive and continuous on $w^+\cap \mathbb {S}^{n-1},$ by Lemma 3.4 (1) and (5.1), we see that $ F_M\rightarrow F_K$ uniformly. By this and

$$\begin{align*}\int_{\mathbb{S}^{n-1}} u F_K(u)du =0,\end{align*}$$

we also have $\tilde F_M\rightarrow F_K$ uniformly, too.

2. If $\partial K$ is smooth and has positive curvature, we can prove in Theorem 5.1 that $K_{M,N,k}\to K$ as $M,N,k\to \infty $ , where $K_{M,N,k}$ is the convex body determined by $h_{M,N,k}$ .

Theorem 5.1 If $\partial K$ is smooth and has positive curvature, then $K_{M,N,k}\to K$ in Hausdorff metric as $M,N,k\to \infty $ .

Proof By Remark 5.1(1), $\tilde F_M\rightarrow F_K$ uniformly. Since $F_K$ is smooth and has positive curvature, when M is sufficiently large, we also have $\tilde F_M>0$ . In addition, for any $g\in C(\mathbb {S}^{n-1}),$

$$\begin{align*}\Big|\int_{\mathbb{S}^{n-1}}g(u) \tilde F_M(u) du - \int_{\mathbb{S}^{n-1}}g(u) F_K(u) du\Big| \to 0, \quad \mathrm{as}~M\to \infty.\end{align*}$$

That is to say, $\tilde F_M(u)du\to F_K(u)du$ weakly. Let $K_M$ be the convex body whose curvature function is $\tilde F_M$ . Since $K_M,K$ are convex bodies such that

$$\begin{align*}\int_{\mathbb{S}^{n-1}} uh_M(u)du = \int_{\mathbb{S}^{n-1}} uh_K(u)du = 0,\end{align*}$$

$K_M\rightarrow K$ in the Hausdorff metric. This is a classical result that can be proved by using Blaschke’s selection theorem (see [28] for reference). The assertion follows from this and Theorem 3.3.

When $f_w=1_{w^+\cap \mathbb {S}^{n-1}}$ , the characteristic function of the open hemisphere, $Q_K$ becomes the partial brightness function (which is a binary function rather than $b_K $ )

$$\begin{align*}R_K(v,w) = \int_{w^+} (v\cdot u)_- dS(K,u).\end{align*}$$

Recalling that the brightness function $b_K$ can only determine an origin-symmetric convex body, for the partial brightness function, we are able to construct the convex body without symmetric assumption. It is reasonable because the change of viewing and illumination directions provides more information.