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arXiv · 2409.07036

On reduced spherical bodies

Michał Musielak
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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 $\fracπ{2}$ on $S^2$. We also consider reduced bodies of thickness at least $\fracπ{2}$, 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 $S^2$. 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 $V$.

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Michał Musielak. 2024-09-11. On reduced spherical bodies. https://arxiv.org/abs/2409.07036

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