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The isoperimetric inequality for convex subsets of the sphere

Farhan Azad, Thomas Beck and Karolina Lokaj

Vol. 16 (2023), No. 2, 343–364
Abstract

We give a new proof of the isoperimetric inequality for geodesically convex subsets of the 2-sphere, with equality only for spherical lunes. Combined with a straightforward approximation argument, this inequality was first proved by Bérard, Besson, and Gallot (Invent. Math. 80:2 (1985), 295–308), who provided a generalization of the Lévy–Gromov isoperimetric inequality. Using a Faber–Krahn theorem, the inequality implies a sharp lower bound on the first Dirichlet–Neumann eigenvalue of domains contained in geodesically convex subsets of the sphere.

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Keywords
isoperimetric inequality, spherical lunes, Faber–Krahn
Mathematical Subject Classification
Primary: 35P15, 53A05
Milestones
Received: 1 May 2022
Revised: 19 May 2022
Accepted: 19 May 2022
Published: 26 May 2023

Communicated by Frank Morgan
Authors
Farhan Azad
Mathematics Department
Fordham University
Bronx, NY
United States
Thomas Beck
Mathematics Department
Fordham University
Bronx, NY
United States
Karolina Lokaj
Mathematics Department
Fordham University
Bronx, NY
United States