Inscribed Polygons that Characterize Inner Product Spaces

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 $\rho =\sqrt{(1+\cos \frac{2k\pi }{2m+1})\mathrm{/}2}$, $(k=1,2,\ldots , m ;:m=1,2,\ldots)$,suchthateverychordof$S$thatsupports$\rho S$touches$\rho S$atitsmiddlepoint\mathrm{.}Ifthisconditionholds,theneverypoint$u\in S$isavertexofaregularpolygonthatisinscribedin$S$ and circumscribed about $\rho S$.