On extreme constant width bodies in ℝ³

2025/01/28 by Hynd, Ryan
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2501.16940

Abstract

We consider the family of constant width bodies in ℝ³ which is convex under Minkowski addition. Extreme shapes cannot be expressed as a nontrivial convex combination of other constant width bodies. We show that each Meissner polyhedra is extreme. We also explain that each constant width body obtained by rotating a symmetric Reuleaux polygon about its axis of symmetry is extreme. In addition, we conjecture a general characterization of all extreme constant width shapes.

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• The density of Meissner polyhedra
• A four-dimensional body of constant width
• Meissner Polyhedra
• Convex bodies of constant width in spaces of constant curvature and the extremal area of Reuleaux triangles
• Peabodies of Constant Width