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 $\rho=\sqrt{(1+\cos\frac{2k\pi}{2m+1})/2}$, $(k=1,2,\ldots , m ;\:m=1,2,\ldots)$, 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 $\rho S$.