Inscribed Polygons that Characterize Inner Product Spaces

Carlos Benítez; Pedro Martín; Diego Yáñez

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arXiv · 1707.09171

Inscribed Polygons that Characterize Inner Product Spaces

Carlos Benítez·Pedro Martín·Diego Yáñez

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

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BibTeX RIS

Carlos Benítez, Pedro Martín, Diego Yáñez. 2017-07-28. Inscribed Polygons that Characterize Inner Product Spaces. https://arxiv.org/abs/1707.09171

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