Convex bodies of constant width in spaces of constant curvature and the extremal area of Reuleaux triangles

2022/03/30 by Károly J. Böröczky, Boroczky, Karoly J., Adam Sagmeister +1
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2203.16636

openalex publication_date 2022/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width D. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.

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