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Sharp Reverse Isoperimetric Inequalities in Nonpositively Curved Cones

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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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Acknowledgements

Stéphane Sabourau would like to thank the Fields Institute and the Department of Mathematics at the University of Toronto for their hospitality while this work was completed.

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Authors and Affiliations

  1. Department of Mathematics, Bar Ilan University, 5290002, Ramat Gan, Israel

    Mikhail G. Katz

  2. Univ Paris Est Creteil, CNRS, LAMA, 94010, Creteil, France

    Stéphane Sabourau

  3. Univ Gustave Eiffel, LAMA, 77447, Marne-la-Vallée, France

    Stéphane Sabourau

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  1. Mikhail G. Katz
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  2. Stéphane Sabourau
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Correspondence to Mikhail G. Katz.

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Partially supported by the ANR project Min-Max (ANR-19-CE40-0014)

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Katz, M.G., Sabourau, S. Sharp Reverse Isoperimetric Inequalities in Nonpositively Curved Cones. J Geom Anal 31, 10510–10520 (2021). https://doi.org/10.1007/s12220-021-00658-5

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  • DOI: https://doi.org/10.1007/s12220-021-00658-5

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