High-dimensional tennis balls

TL;DR

This paper constructs specific continuous odd functions from high-dimensional spheres to lower-dimensional spheres whose epsilon-expansions avoid great circles, revealing new geometric properties and connections to Euclidean subspace conjectures.

Contribution

It introduces a novel construction of functions with controlled epsilon-expansions that do not contain great circles, linking geometric topology with Banach space theory.

Findings

Existence of functions with epsilon-expansions avoiding great circles

Connection to Milman's conjecture on Euclidean subspaces

Implications for high-dimensional geometric topology

Abstract

We show that there exist constants $\alpha,\epsilon>0$ such that for every positive integer $n$ there is a continuous odd function $f:S^m\to S^n$, with $m\geq \alpha n$, such that the $\epsilon$-expansion of the image of $f$ does not contain a great circle. We also show how this result is connected to a conjecture of Vitali Milman about well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to $\ell_2^n$.