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\begin{document}$$d\ge 2$$\end{document} and let K and L be two convex bodies in Rd\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}^d}$$\end{document} such that L⊂intK\documentclass[12pt]{minimal}
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\begin{document}$$L\subset {\text {int}}K$$\end{document} and the boundary of L does not contain a segment. If K and L satisfy the (d+1)\documentclass[12pt]{minimal}
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\begin{document}$$(d+1)$$\end{document}-equichordal property, i.e., for any line l supporting the boundary of L and the points {ζ±}\documentclass[12pt]{minimal}
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\begin{document}$$\{\zeta _{\pm }\}$$\end{document} of the intersection of the boundary of K with l, distd+1(L∩l,ζ+)+distd+1(L∩l,ζ-)=2σd+1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\text {dist}}^{d+1}(L\cap l, \zeta _+)+{\text {dist}}^{d+1}(L\cap l, \zeta _-)=2\sigma ^{d+1} \end{aligned}$$\end{document}holds, where the constant σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} is independent of l, does it follow that K and L are concentric Euclidean balls? We prove that if K and L have C2\documentclass[12pt]{minimal}
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\begin{document}$$C^2$$\end{document}-smooth boundaries and L is a body of revolution, then K and L are concentric Euclidean balls.
