Approximation of spherical convex bodies of constant width π/2

2024/09/01 by Huhe Han, Han, Huhe · 1 citation
Mathematics · #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2409.00596

Abstract

Let C⊂ \mathbbS² be a spherical convex body of constant width τ. It is known that (i) if τ<π/2 then for any ε>0 there exists a spherical convex body C_ε of constant width τ whose boundary consists only of arcs of circles of radius τ such that the Hausdorff distance between C and C_ε is at most ε; (ii) if τ>π/2 then for any ε>0 there exists a spherical convex body C_ε of constant width τ whose boundary consists only of arcs of circles of radius τ-\fracπ2 and great circle arcs such that the Hausdorff distance between C and C_ε is at most ε. In this paper, we present an approximation of the remaining case τ=π/2, that is, if τ=π/2 then for any ε>0 there exists a spherical polytope P_ε of constant width π/2 such that the Hausdorff distance between C and P_ε is at most ε.

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