The intrinsic metric on the unit sphere of a normed space

TL;DR

This paper proves that the intrinsic metric on the unit sphere of a real normed space is strongly equivalent to the standard induced metric, with explicit bounds relating the two.

Contribution

It establishes explicit bounds showing the strong equivalence between the intrinsic and induced metrics on the unit sphere of a normed space.

Findings

Intrinsic metric is strongly equivalent to the induced metric.

Explicit bounds: \[ \|x-y\| \leq d(x,y) \leq \sqrt{2}\pi \|x-y\| \].

Provides a quantitative relationship between two metrics on the sphere.

Abstract

Let $S$ denote the unit sphere of a real normed space. We show that the intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$. Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}\pi\|x-y\|, \] where $d$ denotes the intrinsic metric on $S$.