A Beautiful Inequality by Saint-Venant and Pólya Revisited

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2023-01-13 DOI:10.1080/00029890.2022.2156241

A. Berger

引用次数: 0

Abstract

Abstract In mathematical physics and beyond, one encounters many beautiful inequalities that relate geometric or physical quantities describing the shape or size of a set. Such isoperimetric inequalities often have a long history and many important applications. For instance, the eponymous and most classical of all isoperimetric inequalities was known already in antiquity. It asserts that among all closed planar curves of a given length, the circles with perimeter equal to that length, and only they, enclose the largest area. Though not nearly as well-known, an isoperimetric inequality conjectured by Saint-Venant in the 1850s and first proved by Pólya almost a century later, is also very beautiful and important. By presenting a short proof as well as two simple physical interpretations, this article illustrates why the result deserves to be cherished by every student of applied analysis.

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《美丽的不平等》作者:圣维南和Pólya

在数学物理及其他领域，人们会遇到许多美丽的不等式，它们与描述集合形状或大小的几何或物理量有关。这种等周不等式通常有很长的历史和许多重要的应用。例如，所有等周不等式中最经典的同名不等式在古代就已经为人所知。它断言在给定长度的所有封闭平面曲线中，周长等于该长度的圆，且只有它们包围的面积最大。虽然没有那么出名，但圣维南在19世纪50年代推测出的一个等周不等式，在近一个世纪后由Pólya首次证明，也非常美丽和重要。通过一个简短的证明和两个简单的物理解释，本文说明了为什么这个结果值得每一个应用分析的学生珍惜。

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

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

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