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The
isoperimetric inequality for convex subsets of the sphere
Farhan Azad, Thomas Beck and Karolina Lokaj |
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Vol. 16 (2023), No. 2, 343–364
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Abstract |
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We give a new proof of the isoperimetric inequality for geodesically convex subsets of the -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
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Mathematical Subject Classification
Primary: 35P15, 53A05
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Milestones
Received: 1 May 2022
Revised: 19 May 2022
Accepted: 19 May 2022
Published: 26 May 2023
Communicated by Frank Morgan
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Authors |
| Farhan Azad | |
| Mathematics Department Fordham University Bronx, NY United States |
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| Thomas Beck | |
| Mathematics Department Fordham University Bronx, NY United States |
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| Karolina Lokaj | |
| Mathematics Department Fordham University Bronx, NY United States |
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| © 2023 MSP (Mathematical Sciences Publishers). |
