Non-central sections of convex bodies

TL;DR

This paper investigates a geometric problem about convex bodies and their hyperplane sections, showing positive results in two dimensions under certain conditions and exploring higher-dimensional cases.

Contribution

It provides new insights into the problem of determining convex bodies from hyperplane sections, including positive results in the plane and extensions to higher dimensions.

Findings

In rom the plane, the bodies are equal if the condition holds for two disks.

The problem is studied for various modifications and higher-dimensional analogues.

The paper discusses conditions under which convex bodies are uniquely determined by their hyperplane sections.

Abstract

We study the following open problem, suggested by Barker and Larman. Let $K$ and $L$ be convex bodies in $\mathbb R^n$ ($n\ge 2$) that contain a Euclidean ball $B$ in their interiors. If $\mathrm{vol}_{n-1}(K\cap H) = \mathrm{vol}_{n-1}(L\cap H)$ for every hyperplane $H$ that supports $B$, does it follow that $K=L$? We discuss various modifications of this problem. In particular, we show that in $\mathbb R^2$ the answer is positive if the above condition is true for two disks, none of which is contained in the other. We also study some higher dimensional analogues.