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...
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MDPI AG
2024-10-01
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| author | Youjiang Lin Zhilang Deng |
| author_facet | Youjiang Lin Zhilang Deng |
| author_sort | Youjiang Lin |
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| description | 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. |
| format | Article |
| id | doaj-art-3db79df073a3415caeeae9a76ea5bfc2 |
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| issn | 2075-1680 |
| language | English |
| publishDate | 2024-10-01 |
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| series | Axioms |
| spelling | doaj-art-3db79df073a3415caeeae9a76ea5bfc22025-08-20T02:26:45ZengMDPI AGAxioms2075-16802024-10-01131175110.3390/axioms13110751Spherical Steiner SymmetrizationsYoujiang Lin0Zhilang Deng1School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, ChinaSchool of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, ChinaIn 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.https://www.mdpi.com/2075-1680/13/11/751spherical geometryspherical Steiner symmetrizationouter Minkowski contentspherical distance |
| spellingShingle | Youjiang Lin Zhilang Deng Spherical Steiner Symmetrizations Axioms spherical geometry spherical Steiner symmetrization outer Minkowski content spherical distance |
| title | Spherical Steiner Symmetrizations |
| title_full | Spherical Steiner Symmetrizations |
| title_fullStr | Spherical Steiner Symmetrizations |
| title_full_unstemmed | Spherical Steiner Symmetrizations |
| title_short | Spherical Steiner Symmetrizations |
| title_sort | spherical steiner symmetrizations |
| topic | spherical geometry spherical Steiner symmetrization outer Minkowski content spherical distance |
| url | https://www.mdpi.com/2075-1680/13/11/751 |
| work_keys_str_mv | AT youjianglin sphericalsteinersymmetrizations AT zhilangdeng sphericalsteinersymmetrizations |