On an Equichordal Property of a Pair of Convex Bodies

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