Describing the equality conditions of the Alexandrov–Fenchel inequality [Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81] order polytopes and employs poset theoretic technology.

We prove a weak version of the cross-product conjecture: $\textrm {F}(k+1,\ell ) \hskip .06cm \textrm {F}(k,\ell +1) \ge (\frac 12+\varepsilon ) \hskip .06cm \textrm {F}(k,\ell ) \hskip .06cm \textrm {F}(k+1,\ell +1)$, where $\textrm {F}(k,\ell )$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are k and $\ell $ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the generalized cross-product conjecture. The proofs use geometric inequalities for mixed volumes and combinatorics of words.

We investigate the convexity of the radial sum of two convex bodies containing the origin. Generally, the radial sum of two convex bodies containing the origin is not convex. We show that the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a pair of convex bodies containing the origin whose radial sum is convex.

We also investigate the convexity of the intersection body of a convex body containing the origin. Generally, the intersection body of a convex body containing the origin is not convex. Busemann’s theorem states that the intersection body of any centered convex body is convex. We are interested in how to construct convex intersection bodies from convex bodies without any symmetry (especially, central symmetry). We show that the intersection body of the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a convex body with no symmetries whose intersection body is convex.

We investigate the weighted $L_p$ affine surface areas which appear in the recently established $L_p$ Steiner formula of the $L_p$ Brunn–Minkowski theory. We show that they are valuations on the set of convex bodies and prove isoperimetric inequalities for them. We show that they are related to f divergences of the cone measures of the convex body and its polar, namely the Kullback–Leibler divergence and the Rényi divergence.

Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

In this paper, we consider the family of nth degree polynomials whose coefficients form a log-convex sequence (up to binomial weights), and investigate their roots. We study, among others, the structure of the set of roots of such polynomials, showing that it is a closed convex cone in the upper half-plane, which covers its interior when n tends to infinity, and giving its precise description for every $n\in \mathbb {N}$ , $n\geq 2$ . Dual Steiner polynomials of star bodies are a particular case of them, and so we derive, as a consequence, further properties for their roots.

We shall prove a convergence result relative to sequences of Minkowski symmetrals of general compact sets. In particular, we investigate the case when this process is induced by sequences of subspaces whose elements belong to a finite family, following the path marked by Klain in Klain (2012, Advances in Applied Mathematics 48, 340–353), and the generalizations in Bianchi et al. (2019, Convergence of symmetrization processes, preprint) and Bianchi et al. (2012, Indiana University Mathematics Journal 61, 1695–1710). We prove an analogous result for fiber symmetrization of a specific class of compact sets. The idempotency for symmetrizations of this family of sets is investigated, leading to a simple generalization of a result from Klartag (2004, Geometric and Functional Analysis 14, 1322–1338) regarding the approximation of a ball through a finite number of symmetrizations, and generalizing an approximation result in Fradelizi, Làngi and Zvavitch (2020, Volume of the Minkowski sums of star-shaped sets, preprint).

A. R. Martínez Fernández obtained upper bounds for quermassintegrals of the p-inner parallel bodies: an extension of the classical inner parallel body to the $L_p$ -Brunn-Minkowski theory. In this paper, we establish (sharp) upper and lower bounds for quermassintegrals of p-inner parallel bodies. Moreover, the sufficient and necessary conditions of the equality case for the main inequality are obtained, which characterize the so-called tangential bodies.

In the present paper, we first introduce the concepts of the Lpq-capacity measure and Lp mixed q-capacity and then prove some geometric properties of Lpq-capacity measure and a Lp Minkowski inequality for the q-capacity for any fixed p ⩾ 1 and q > n. As an application of the Lp Minkowski inequality mentioned above, we establish a Hadamard variational formula for the q-capacity under p-sum for any fixed p ⩾ 1 and q > n, which extends results of Akman et al. (Adv. Calc. Var. (in press)). With the Hadamard variational formula, variational method and Lp Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the q-capacity which extends some beautiful results of Jerison (1996, Acta Math.176, 1–47), Colesanti et al. (2015, Adv. Math.285, 1511–588), Akman et al. (Mem. Amer. Math. Soc. (in press)) and Akman et al. (Adv. Calc. Var. (in press)). It is worth mentioning that our proof of Hadamard variational formula is based on Lp Minkowski inequality rather than the direct argument which was adopted by Akman (Adv. Calc. Var. (in press)). Moreover, as a consequence of Lp Minkowski inequality for q-capacity, we get an interesting isoperimetric inequality for q-capacity.

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\], where ɛ:= {x ∈ ℝd : x┬ (A┬A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p∫Ad,k |ɛ ∩ E|p+d+1μd,k(dE) = |ɛ|k+1∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$. Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

We generalize to the anisotropic case some classical and recent results on the (n – 1)-Minkowski content of rectifiable sets in ℝn, and on the outer Minkowski content of subsets of ℝn. In particular, a general formula for the anisotropic outer Minkowski content is provided; it applies to a wide class of sets that are stable under finite unions.

Let K ⊂ ℝN be any convex body containing the origin. A measurable set G ⊂ ℝN with finite and positive Lebesgue measure is said to be K-dense if, for any fixed r > 0, the measure of G ⋂ (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). In a previous work, we proved for the case N = 2 that if G is K-dense, then both G and K must be homothetic to the same ellipse. Here, we completely characterize K-dense sets in ℝN: if G is K-dense, then both G and K must be homothetic to the same ellipsoid. Our proof, which builds upon results obtained in our previous work, relies on an asymptotic formula for the measure of G ⋂ (x + rK) for large values of the parameter r and a classical characterization of ellipsoids due to Petty.

In this note we present inequalities relating the successive minima of an $o$-symmetric convex body and the successive inner and outer radii of the body. These inequalities join known inequalities involving only either the successive minima or the successive radii.

In the Euclidean plane ${{\mathbb{R}}^{2}}$, we define the Minkowski difference $\mathcal{K}-\mathcal{L}$ of two arbitrary convex bodies $\mathcal{K},\mathcal{L}$ as a rectifiable closed curve ${{\mathcal{H}}_{h}}\subset {{\mathbb{R}}^{2}}$ that is determined by the difference $h={{h}_{K}}-{{h}_{\mathcal{L}}}$ of their support functions. This curve ${{\mathcal{H}}_{h}}$ is called the hedgehog with support function $h$. More generally, the object of hedgehog theory is to study the Brunn–Minkowski theory in the vector space of Minkowski differences of arbitrary convex bodies of Euclidean space ${{\mathbb{R}}^{n+1}}$, defined as (possibly singular and self-intersecting) hypersurfaces of ${{\mathbb{R}}^{n+1}}$. Hedgehog theory is useful for: (i) studying convex bodies by splitting them into a sum in order to reveal their structure; (ii) converting analytical problems into geometrical ones by considering certain real functions as support functions. The purpose of this paper is to give a detailed study of plane hedgehogs, which constitute the basis of the theory. In particular: (i) we study their length measures and solve the extension of the Christoffel–Minkowski problem to plane hedgehogs; (ii) we characterize support functions of plane convex bodies among support functions of plane hedgehogs and support functions of plane hedgehogs among continuous functions; (iii) we study the mixed area of hedgehogs in ${{\mathbb{R}}^{2}}$ and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.

Intrinsic volumes are key functionals in convex geometry and have also appeared in several stochastic settings. Here we relate them to questions of regularity in Gaussian processes with regard to Itô–Nisio oscillation and metrization of GB/GC indexing sets. Various bounds and estimates are presented, and questions for further investigation are suggested. From alternate points of view, much of the discussion can be interpreted in terms of (i) random sets and (ii) properties of (deterministic) infinite-dimensional convex bodies.

In generalization of the well-known formulae for quermass densities of stationary and isotropic Boolean models, we prove corresponding results for densities of mixed volumes in the stationary situation and show how they can be used to determine the intensity of non-isotropic Boolean models Z in d-dimensional space for d = 2, 3, 4. We then consider non-stationary Boolean models and extend results of Fallert on quermass densities to densities of mixed volumes. In particular, we present explicit formulae for a planar inhomogeneous Boolean model with circular grains.

Methods of estimation of the oriented direction distribution (i.e. the distribution of unit outer normals over the boundary) of a planar set from the convex ring are proposed. The methods are based on an estimation of the area dilation of the investigated set by chosen test sets.

We give a proof of the Steiner formula based on the theory of random convex bodies. In particular, we make use of laws of large numbers for both random volumes and random convex bodies themselves.