Distortion from spheres into Euclidean spaces
James Dibble
TL;DR
This work establishes parity-dependent universal lower bounds on the distortion of any map $f:S^n_r \to \mathbb{R}^n$, refining the classical Borsuk--Ulam constraint. The authors first prove the one-dimensional case via a combinatorial construction and then extend to higher dimensions by employing Granas' set-valued fixed-point theorem, recasting the problem in terms of convex hulls and intersecting simplices. A key geometric ingredient is a sharp bound on the maximal distance between vertices of two intersecting simplices with bounded edge length, achieved by orthogonal regular simplices of dimensions $\lfloor n/2 \rfloor$ and $\lceil n/2 \rceil$. Consequently, the main theorem provides explicit lower bounds on distortion: $\mathrm{dist}(f) \ge \frac{\pi r}{1 + \sqrt{1 - \frac{2}{n+2}}}$ for even $n$ and $\mathrm{dist}(f) \ge \frac{\pi r}{1 + \sqrt{1 - \frac{2(n+2)}{(n+1)(n+3)}}}$ for odd $n$, with a universal bound when target dimension $m \le n$ giving $\mathrm{dist}(f) > \frac{\pi r}{2}$.
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 $\displaystyle \frac{πr}{1 + \sqrt{1 - \frac{2}{n+2}}}$ if $n$ is even and $\displaystyle \frac{πr}{1 + \sqrt{1 - \frac{2(n+2)}{(n+1)(n+3)}}}$ 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.