Sharp reverse isoperimetric inequalities in nonpositively curved cones

Mikhail G. Katz; Stephane Sabourau

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arXiv · 2103.04661

Sharp reverse isoperimetric inequalities in nonpositively curved cones

Mikhail G. Katz·Stephane Sabourau

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Abstract

We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least $2\pi$ have minimal area among all nonpositively curved disks of the same perimeter and the same total curvature; (2) geodesic triangles in a Euclidean (resp. hyperbolic) cone of angle at least $2\pi$ have minimal area among all nonpositively curved geodesic triangles (resp. all geodesic triangles of curvature at most $-1$) with the same side lengths and angles.

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Mikhail G. Katz, Stephane Sabourau. 2021-03-08. Sharp reverse isoperimetric inequalities in nonpositively curved cones. https://arxiv.org/abs/2103.04661

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