arXiv subjects

Michał Musielak

Publications and source records attributed to Michał Musielak.

5 recordsLinked to original sources

arXiv2024

On reduced spherical bodies

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 $V$.

math.MG↗

arXiv2018

Diameter of reduced spherical convex bodies

The intersection $L$ of two different non-opposite hemispheres of the unit sphere $S^2$ is called a lune. By $Δ(L)$ we denote the distance of the centers of the semicircles bounding $L$. By the thickness $Δ(C)$ of a convex body $C \subset S^2$ we mean the minimal value of $Δ(L)$ over all lunes $L \supset C$. We call a convex body $R\subset S^2$ reduced provided $Δ(Z) < Δ(R)$ for every convex body $Z$ being a proper subset of $R$. Our aim is to estimate the diameter of $R$, where $Δ(R) < \fracπ{2}$, in terms of its thickness.

math.MG↗

arXiv2018

Covering a reduced spherical body by a disk

In this paper, the following two theorems are proved: $(1)$ every spherical convex body $W$ of constant width $Δ(W) \geq \fracπ{2}$ may be covered by a disk of radius $Δ(W) + \arcsin \left( \frac{2\sqrt{3}}{3} \cdot \cos \frac{Δ(W)}{2}\right) - \fracπ{2}$; $(2)$ every reduced spherical convex body $R$ of thickness $Δ(R)<\fracπ{2}$ may be covered by a disk of radius $\arctan \left( \sqrt{2} \cdot \tan \frac{Δ(R)}{2}\right)$.

math.MG↗

arXiv2018

Spherical bodies of constant width

The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of a $d$-dimensional sphere $S^d$ is called a lune. By the thickness of $L$ we mean the distance of the centers of the $(d-1)$-dimensional hemispheres bounding $L$. For a hemisphere $G$ supporting a %spherical convex body $C \subset S^d$ we define ${\rm width}_G(C)$ as the thickness of the narrowest lune or lunes of the form $G \cap H$ containing $C$. If ${\rm width}_G(C) =w$ for every hemisphere $G$ supporting $C$, we say that $C$ is a body of constant width $w$. We present properties of these bodies. In particular, we prove that the diameter of any spherical body $C$ of constant width $w$ on $S^d$ is $w$, and that if $w < \fracπ{2}$, then $C$ is strictly convex. Moreover, we are checking when spherical bodies of constant width and constant diameter coincide.

math.MG↗

arXiv2016

Reduced Spherical Convex Bodies

The aim of this paper is to present some properties of reduced spherical convex bodies on the two-dimensional sphere $S^2$. The intersection of two different non-opposite hemispheres is called a lune. By its thickness we mean the distance of the centers of the two semicircles bounding it. The thickness $Δ(C)$ of $C$ is the minimum thickness of a lune containing $C$. We say that a spherical convex body $R$ is reduced if $Δ(Z) < Δ(R)$ for every spherical convex body $Z \subset R$ different from $R$. Our main theorem permits to describe the shape of reduced bodies of thickness below $\fracπ{2}$. It implies a number of corollaries. In particular, we estimate the diameter of reduced spherical bodies in terms of their thickness. Reduced bodies of thickness at least $\fracπ{2}$ have constant width. Spherical convex bodies of constant width below $\fracπ{2}$ are strictly convex.

math.MG↗