Inscribed Polygons that Characterize Inner Product Spaces

TL;DR

This paper characterizes inner product spaces among normed spaces using inscribed polygons and chord conditions on the unit sphere, providing a geometric criterion for inner product structure.

Contribution

It introduces a novel geometric characterization of inner product spaces based on inscribed polygons and chord support conditions on the unit sphere.

Findings

Inner product spaces are characterized by specific inscribed polygon conditions.

The existence of certain inscribed polygons implies the space is an inner product space.

Every point on the sphere forms a vertex of a regular polygon inscribed in the sphere.

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 $\rho S$.