Inscribed Polygons that Characterize Inner Product Spaces

Abstract

Let $X$ be a real normed space with unit sphere S. We prove that $X$ is an inner product space if and only if there exists a real number $\rho =\sqrt{(1+\cos \frac{2k\pi }{2m+1})/2}$, $(k=1,2,\dots ,m;m=1,2,\dots )$, such that every chord of $S$ that supports $\rho S$ touches $\rho S$ at its middle point. If this condition holds, then every point $u\in S$ is a vertex of a regular polygon that is inscribed in $S$ and circumscribed about $\rho S$.

12 pages, 2 figures