A characterization of the sphere (Q5943220)

Let \(\varphi(t)\), where \(t\in [a,b]\), be a closed piecewise \({\mathcal C}'\)-curve contained in the interior of a ball in the Euclidean \(n\)-space. Denote by \(\varphi_+(t)\) and by \(\varphi_-(t)\) the distances of the point \(\varphi(t)\) in the directions \(\varphi'(t)\) and \(-\varphi'(t)\), respectively, from the sphere. Moreover, assume that \(\|\varphi'(t) \|=1\) for almost all \(t\in [a,b]\). The authors show that \(\int^b_a \varphi_+(t)dt =\int^b_a \varphi_-(t)dt\). They also prove that this property characterizes balls among convex bodies.