Convex polygons and the isoperimetric problem in simply connected space forms M_κ²

2024/08/24 by A. R. Aithal, Anisa M. H. Chorwadwala, Aithal, A R +1
Computer Science · Mathematics · #53A04 #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications #Optimization and Control (math.OC) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2408.13565

openalex publication_date 2024/08/24 · openalex created_date 2024/09/21 · openalex updated_date 2026/07/28

Abstract

In this article, we prove that there exists a unique perimeter minimizer among all piecewise smooth simple closed curves in M_κ² enclosing area A > 0 (A ≤ 2π if κ = 1), and it is a circle in M_κ² of radius AS_κ ( \dfrac √( A ( 4 π - κ A ) ) 2 π ), where AS_κ(t) := t if κ = 0, arcsin(t) if κ = 1, sinh^-1(t) if κ =-1. We also prove the isoperimetric inequality for M_κ². We give an elementary geometric proof which is uniform for all three simply connected space forms.

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