Spherical Steiner Symmetrizations

Spherical Steiner Symmetrizations

In this paper, we primarily investigate and establish several properties of spherical Steiner symmetrizations, along with the isoperimetric property of the spherical cap in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics...

Full description

Saved in:
Bibliographic Details
Main Authors: Youjiang Lin, Zhilang Deng
Format: Article
Language:English
Published: MDPI AG 2024-10-01
Series:Axioms
Subjects:
spherical geometry
spherical Steiner symmetrization
outer Minkowski content
spherical distance
Online Access:https://www.mdpi.com/2075-1680/13/11/751
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In this paper, we primarily investigate and establish several properties of spherical Steiner symmetrizations, along with the isoperimetric property of the spherical cap in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">S</mi><mi>n</mi></msup></semantics></math></inline-formula>. Specifically, we study the monotonically decreasing property of the measure of the symmetric difference of two spherical compact sets, the monotonically decreasing property of the spherical diameter of a spherical compact set, the convergence of iterative spherical Steiner symmetrizations, and so on. In particular, we prove that the sequence of iterative spherical Steiner symmetrizations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">S</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula>, which follow sequences selected from a finite set of directions, converges to a spherical cap with the same measure as <i>K</i>, extending the result from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">S</mi><mi>n</mi></msup></semantics></math></inline-formula> on Steiner symmetrizations. It provides us with valuable insights for studying the relevant applications and conclusions of spherical Steiner symmetrizations.
ISSN:2075-1680

Similar Items