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,

$$\begin{aligned} {\text {dist}}^{d+1}(L\cap l, \zeta _+)+{\text {dist}}^{d+1}(L\cap l, \zeta _-)=2\sigma ^{d+1} \end{aligned}$$

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.