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arXiv · 1510.07442

The intrinsic metric on the unit sphere of a normed space

Miek Messerschmidt·Marten Wortel
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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 $S$.

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Miek Messerschmidt, Marten Wortel. 2015-10-26. The intrinsic metric on the unit sphere of a normed space. https://arxiv.org/abs/1510.07442

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