{"title":"关于Borsuk–Ulam定理和凸集","authors":"M. C. Crabb","doi":"10.1112/mtk.12186","DOIUrl":null,"url":null,"abstract":"<p>The Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem of Adams, Bush and Frick [1] that if <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>:</mo>\n <msup>\n <mi>S</mi>\n <mn>1</mn>\n </msup>\n <mo>→</mo>\n <msup>\n <mi>R</mi>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n </mrow>\n <annotation>$f: S^1\\rightarrow {\\mathbb {R}}^{2k+1}$</annotation>\n </semantics></math> is a continuous function on the unit circle <i>S</i><sup>1</sup> in <math>\n <semantics>\n <mi>C</mi>\n <annotation>${\\mathbb {C}}$</annotation>\n </semantics></math> such that <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>(</mo>\n <mo>−</mo>\n <mi>z</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mo>−</mo>\n <mi>f</mi>\n <mo>(</mo>\n <mi>z</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$f(-z)=-f(z)$</annotation>\n </semantics></math> for all <math>\n <semantics>\n <mrow>\n <mi>z</mi>\n <mo>∈</mo>\n <msup>\n <mi>S</mi>\n <mn>1</mn>\n </msup>\n </mrow>\n <annotation>$z\\in S^1$</annotation>\n </semantics></math>, then there is a finite subset <i>X</i> of <i>S</i><sup>1</sup> of diameter at most <math>\n <semantics>\n <mrow>\n <mi>π</mi>\n <mo>−</mo>\n <mi>π</mi>\n <mo>/</mo>\n <mo>(</mo>\n <mn>2</mn>\n <mi>k</mi>\n <mo>+</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$\\pi -\\pi /(2k+1)$</annotation>\n </semantics></math> (in the standard metric in which the circle has circumference of length 2π) such the convex hull of <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>(</mo>\n <mi>X</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$f(X)$</annotation>\n </semantics></math> contains <math>\n <semantics>\n <mrow>\n <mn>0</mn>\n <mo>∈</mo>\n <msup>\n <mi>R</mi>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n </mrow>\n <annotation>$0\\in {\\mathbb {R}}^{2k+1}$</annotation>\n </semantics></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"On Borsuk–Ulam theorems and convex sets\",\"authors\":\"M. C. Crabb\",\"doi\":\"10.1112/mtk.12186\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem of Adams, Bush and Frick [1] that if <math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n <mo>:</mo>\\n <msup>\\n <mi>S</mi>\\n <mn>1</mn>\\n </msup>\\n <mo>→</mo>\\n <msup>\\n <mi>R</mi>\\n <mrow>\\n <mn>2</mn>\\n <mi>k</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$f: S^1\\\\rightarrow {\\\\mathbb {R}}^{2k+1}$</annotation>\\n </semantics></math> is a continuous function on the unit circle <i>S</i><sup>1</sup> in <math>\\n <semantics>\\n <mi>C</mi>\\n <annotation>${\\\\mathbb {C}}$</annotation>\\n </semantics></math> such that <math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n <mo>(</mo>\\n <mo>−</mo>\\n <mi>z</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mo>−</mo>\\n <mi>f</mi>\\n <mo>(</mo>\\n <mi>z</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$f(-z)=-f(z)$</annotation>\\n </semantics></math> for all <math>\\n <semantics>\\n <mrow>\\n <mi>z</mi>\\n <mo>∈</mo>\\n <msup>\\n <mi>S</mi>\\n <mn>1</mn>\\n </msup>\\n </mrow>\\n <annotation>$z\\\\in S^1$</annotation>\\n </semantics></math>, then there is a finite subset <i>X</i> of <i>S</i><sup>1</sup> of diameter at most <math>\\n <semantics>\\n <mrow>\\n <mi>π</mi>\\n <mo>−</mo>\\n <mi>π</mi>\\n <mo>/</mo>\\n <mo>(</mo>\\n <mn>2</mn>\\n <mi>k</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\pi -\\\\pi /(2k+1)$</annotation>\\n </semantics></math> (in the standard metric in which the circle has circumference of length 2π) such the convex hull of <math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$f(X)$</annotation>\\n </semantics></math> contains <math>\\n <semantics>\\n <mrow>\\n <mn>0</mn>\\n <mo>∈</mo>\\n <msup>\\n <mi>R</mi>\\n <mrow>\\n <mn>2</mn>\\n <mi>k</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$0\\\\in {\\\\mathbb {R}}^{2k+1}$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":18463,\"journal\":{\"name\":\"Mathematika\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-02-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematika\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12186\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12186","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem of Adams, Bush and Frick [1] that if is a continuous function on the unit circle S1 in such that for all , then there is a finite subset X of S1 of diameter at most (in the standard metric in which the circle has circumference of length 2π) such the convex hull of contains .
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.
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