Sharp reverse isoperimetric inequalities in nonpositively curved cones

2021/03/08 by Katz, Mikhail G., Sabourau, Stephane
#49Q10 #53C21 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2103.04661

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 2π 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 2π 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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