Search

We show that if d≥4 is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.

Let $K$ be a convex body with boundary of class ${\mathcal{C}}^{\geq 4}$. We prove that, for a sufficiently small $\delta$, the floating body $K_{\delta}$ is homothetic to $K$ if and only if $K$ is an ellipsoid. The proof relies on properties of the affine curvature flow.

The covariogram of a convex body $K$ provides the volumes of the intersections of $K$ with all its possible translates. Matheron conjectured in 1986 that this information determines $K$ among all convex bodies, up to certain known ambiguities. It is proved that this is the case if $K\subset{\mathbb R}^2$ is not ${\rm C}^1$, or it is not strictly convex, or its boundary contains two arbitrarily small ${\rm C}^2$ open portions ‘on opposite sides’. Examples are also constructed that show that this conjecture is false in ${\mathbb R}^n$ for any $n\geq 4$.

Search Results

Refine search

Refine search

Actions for selected content:

3 results

NON-UNIQUENESS OF CONVEX BODIES WITH PRESCRIBED VOLUMES OF SECTIONS AND PROJECTIONS

THE FLOATING BODY PROBLEM

MATHERON'S CONJECTURE FOR THE COVARIOGRAM PROBLEM

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

3 results

NON-UNIQUENESS OF CONVEX BODIES WITH PRESCRIBED VOLUMES OF SECTIONS AND PROJECTIONS

THE FLOATING BODY PROBLEM

MATHERON'S CONJECTURE FOR THE COVARIOGRAM PROBLEM