On Borsuk–Ulam theorems and convex sets

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematika Pub Date : 2023-02-11 DOI:10.1112/mtk.12186

M. C. Crabb

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引用次数: 5

Abstract

The Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem of Adams, Bush and Frick [1] that if $f : S^{1} \to R^{2 k + 1}$ is a continuous function on the unit circle S¹ in $C$ such that $f (- z) = - f (z)$ for all $z \in S^{1}$ , then there is a finite subset X of S¹ of diameter at most $π - π / (2 k + 1)$ (in the standard metric in which the circle has circumference of length 2π) such the convex hull of $f (X)$ contains $0 \in R^{2 k + 1}$ .

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关于Borsuk–Ulam定理和凸集

中间值定理用于给出Adams、Bush和Frick[1]的Borsuk–Ulam定理的初等证明，如果f:S1→R2k+1$f:S^1\rightarrow｛\mathbb｛R｝｝^｛2k+1｝$是C$中单位圆S1上的连续函数，使得对于S^1$中的所有z∈S1$z\，则S1的一个有限子集X的直径至多为π-π/（2k+1）$\pi-\pi/（2k+1。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematika MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.