A Borsuk--Ulam theorem for well separated maps

M. C. Crabb

A Borsuk--Ulam theorem for well separated maps

M. C. Crabb

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 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.

A Borsuk--Ulam theorem for well separated maps

Abstract

Suppose that

are

(

) continuous functions defined on the unit sphere in a Euclidean vector space

of dimension

satisfying

for all

. The classical Borsuk-Ulam theorem asserts that the image of the map

contains

. 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

-cube.

Paper Structure (2 theorems, 5 equations)

This paper contains 2 theorems, 5 equations.

Key Result

Theorem 1

Let $V$ be a Euclidean vector space of dimension $m+1>1$. Suppose that $f_1,\ldots , f_m :S(V)\to {\mathbb R}$ are continuous functions such that $f_i(-v)=-f_i(v)$ for all $v\in S(V)$, $i=1,\ldots ,m$, and satisfying the condition that the open subset of $S(V)$ can be written as the disjoint union $\Omega =\Omega_+\sqcup\Omega_-$ of two open subsets which are interchanged by the antipodal involut

Theorems & Definitions (8)

Theorem 1
proof
Remark 2
Example 3
proof
Corollary 4
proof
Example 5