A Beautiful Inequality by Saint-Venant and Pólya Revisited

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.

MSC:

• 35A23
• 26D10
• 97M50
• 35J25
• 74B05
• 76D03

Acknowledgments

The author was supported by an Nserc Discovery Grant. He is grateful to T. Hillen and G. Huisken for insightful communications. He also wishes to thank two anonymous referees who brought the recent preprint [9] to his attention and made suggestions that improved the exposition.

Additional information

Notes on contributors

Arno Berger

ARNO BERGER is a professor of mathematics at the University of Alberta. He is interested primarily in dynamical systems, analysis, and probability theory. Before settling in Canada, he held academic positions in Austria and New Zealand, and spent extended research visits to the US, UK, and Germany. He received his Ph.D. from the Technische Universität Wien in 1997.