Mixed volumes and the Blaschke-Lebesgue theorem

2023/11/24 by Beniamin Bogosel, Bogosel, Beniamin
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2311.14618

openalex publication_date 2023/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The mixed area of a Reuleaux polygon and its symmetric with respect to the origin is expressed in terms of the mixed area of two explicit polygons. This gives a geometric explanation of a classical proof due to Chakerian. Mixed areas and volumes are also used to reformulate the minimization of the volume under constant width constraint as isoperimetric problems. In the two dimensional case, the equivalent formulation is solved, providing another proof of the Blaschke-Lebesgue theorem. In the three dimensional case the proposed relaxed formulation involves the mean width, the area and inclusion constraints.

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