On an equichordal property of a pair of convex bodies

Abstract

Let $d\ge 2$ and let $K$ and $L$ be two convex bodies in ${\mathbb{R}}^d$ such that $L\subset \text{int}K$ 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 $l$, $${\text{dist}}^{d+1}(L\cap l,{\zeta }_{+})+{\text{dist}}^{d+1}(L\cap l,{\zeta }_{-})=2{\sigma }^{d+1}$$ 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.

9 figures, 20 pages