High-dimensional tennis balls

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. This result is motivated by a conjecture of Vitali Milman about well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to \(\ell_2^n\).Mathematics Subject Classifications: 46B09, 60C05Keywords: Anti-Ramsey, antipodal subsphere