Unique local determination of convex bodies

TL;DR

This paper investigates the local uniqueness of convex bodies determined by sections or caps, providing positive results for small perturbations of smooth convex bodies and extending to various geometric measures.

Contribution

It offers new local uniqueness results for convex bodies based on sections or caps, including generalizations to quermassintegrals and lower-dimensional affine sections.

Findings

Unique local determination of convex bodies from section volumes.

Extension to quermassintegrals of various dimensions.

Use of specific affine subspaces for determination.

Abstract

Barker and Larman asked the following. Let $K' \subset {\Bbb{R}}^d$ be a convex body, whose interior contains a given convex body $K \subset {\Bbb{R}}^d$, and let, for all supporting hyperplanes $H$ of $K$, the $(d-1)$-volumes of the intersections $K' \cap H$ be given. Is $K'$ then uniquely determined? Yaskin and Zhang asked the analogous question when, for all supporting hyperplanes $H$ of $K$, the $d$-volumes of the "caps" cut off from $K'$ by $H$ are given. We give local positive answers to both of these questions, for small $C^2$-perturbations of $K$, provided the boundary of $K$ is $C^2_+$. In both cases, $(d-1)$-volumes or $d$-volumes can be replaced by $k$-dimensional quermassintegrals for $1 \le k \le d-1$ or for $1 \le k \le d$, respectively. Moreover, in the first case we can admit, rather than hyperplane sections, sections by $l$-dimensional affine planes, where $1 \le k \le l \le d-1$. In fact, here not all $l$-dimensional affine subspaces are needed, but only a small subset of them (actually, a $(d-1)$-manifold), for unique local determination of $K'$.