On reduced spherical bodies

2024/09/11 by Michał Musielak, Musielak, Michał
Engineering · Mathematics · #FOS: Mathematics #Material Science and Thermodynamics #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2409.07036

openalex publication_date 2024/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This thesis consists of five papers about reduced spherical convex bodies and in particular spherical bodies of constant width on the d-dimensional sphere S^d. In paper I we present some facts describing the shape of reduced bodies of thickness under \fracπ2 on S². We also consider reduced bodies of thickness at least \fracπ2, which appear to be of constant width. Paper II focuses on bodies of constant width on S^d. We present the properties of these bodies and in particular we discuss conections between notions of constant width and of constant diameter. In paper III we estimate the diameter of a reduced convex body. The main theme of paper IV is estimating the radius of the smallest disk that covers a reduced convex body on S². The result of paper V is showing that every spherical reduced polygon V is contained in a disk of radius equal to the thickness of this body centered at a boundary point of V.

Related

• Reduced Spherical Convex Bodies
• Spherical geometry -- a survey on width and thickness of convex bodies
• Diameter of reduced spherical convex bodies
• Three characterizations of reducedness of spherical convex polygons
• Covering a reduced spherical body by a disk