A new characterization of convex plates of constant width (Q1263814)

It is shown that a convex figure \(D\subset {\mathbb{R}}^ 2\) of diameter 1 is of constant width 1 if and only if any two perpendicular chords with non- empty intersection have total length \(\geq 1\). Moreover, if D has constant width 1, there is a strict inequality for non-degenerate chords. One of both implications is proved for higher dimensions, too: If any n mutually perpendicular chords of a convex body \(K\subset {\mathbb{R}}^ n\), carrying a common point, have total length \(\geq 1\) (for non-degenerate chords), then K is of constant width 1. There remains the open problem, whether for \(n\geq 3\) the contrary implication holds.