A Borsuk--Ulam theorem for well separated maps

M. C. Crabb

paper

A Borsuk--Ulam theorem for well separated maps

arXiv:2401.02209

Abstract

Suppose that $f_{1}, \dots, f_{m} : S (V) \to R$ are $m$ ( $\geq 1$ ) continuous functions defined on the unit sphere in a Euclidean vector space $V$ of dimension $m + 1$ satisfying $f_{i} (- v) = - f_{i} (v)$ for all $v \in S (V)$ . The classical Borsuk-Ulam theorem asserts that the image of the map $(f_{1}, \dots, f_{m}) : S (V) \to R^{m}$ contains $0 = (0, \dots, 0)$ . Pursuing ideas in papers of Bárány, Hubard and Jéronimo (2008) and Frick and Wellner (2023), we show that a certain separation property will guarantee that the image contains an $m$ -cube.