On an equichordal property of a pair of convex bodies

TL;DR

This paper investigates a geometric property called the $(d+1)$-equichordal property for convex bodies and proves that under certain smoothness and symmetry conditions, the bodies must be concentric Euclidean balls.

Contribution

The paper establishes that convex bodies with the $(d+1)$-equichordal property, smooth boundaries, and rotational symmetry are necessarily concentric Euclidean balls, extending understanding of equichordal configurations.

Findings

Bodies with the property are Euclidean balls under smoothness and symmetry conditions.

The $(d+1)$-equichordal property characterizes concentric Euclidean balls.

The result applies to bodies with $C^2$-smooth boundaries and rotational symmetry.

Abstract

Let $d\ge 2$ and let $K$ and $L$ be two convex bodies in ${\mathbb R^d}$ such that $L\subset \textrm{int}\,K$ and the boundary of $L$ does not contain a segment. If $K$ and $L$ satisfy the $(d+1)$-equichordal property, i.e., for any line $l$ supporting the boundary of $L$ and the points $\{\zeta_{\pm}\}$ of the intersection of the boundary of $K$ with $l$, $$ \textrm{dist}^{d+1}(L\cap l, \zeta_+)+\textrm{dist}^{d+1}(L\cap l, \zeta_-)=2\sigma^{d+1} $$ holds, where the constant $\sigma$ is independent of $l$, does it follow that $K$ and $L$ are concentric Euclidean balls? We prove that if $K$ and $L$ have $C^2$-smooth boundaries and $L$ is a body of revolution, then $K$ and $L$ are concentric Euclidean balls.