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arXiv · 2504.02276

Distortion from spheres into Euclidean spaces

James Dibble
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Abstract

Any function from a round $n$-dimensional sphere of radius $r$ into $n$-dimensional Euclidean space must distort the metric additively by at least $\displaystyle \frac{\pi r}{1 + \sqrt{1 - \frac{2}{n+2}}}$ if $n$ is even and $\displaystyle \frac{\pi r}{1 + \sqrt{1 - \frac{2(n+2)}{(n+1)(n+3)}}}$ if $n$ is odd. This is proved using a fixed-point theorem of Granas that generalizes the classical theorem of Borsuk-Ulam to set-valued functions.

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James Dibble. 2025-04-03. Distortion from spheres into Euclidean spaces. https://arxiv.org/abs/2504.02276

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