A Borsuk--Ulam theorem for well separated maps

M. C. Crabb

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arXiv · 2401.02209

A Borsuk--Ulam theorem for well separated maps

M. C. Crabb

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Abstract

Suppose that $f_1,\ldots ,f_m : S(V)\to R$ 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 $(f_1,\ldots ,f_m) :S(V)\to R^m$ contains $0=(0,\ldots ,0)$. Pursuing ideas in papers of B\'ar\'any, Hubard and J\'eronimo (2008) and Frick and Wellner (2023), we show that a certain separation property will guarantee that the image contains an $m$-cube.

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M. C. Crabb. 2024-01-04. A Borsuk--Ulam theorem for well separated maps. https://arxiv.org/abs/2401.02209

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