A Cap Covering Theorem
A Cap Covering Theorem
复制标题
上限覆盖定理
作者：
A. Polyanskii
单位 d 球 S 上的球面半径 α 的上限是距球体上给定点的球面距离 α 内的点的集合。令 ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal F}$$end{document} 是位于 S 上的有限大写集。我们证明如果没有通过 S 中心的超平面将 ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal F}$$end{document} 一分为二ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal F}$$end{document}中不相交的非空子集，然后有是一个半径上限，等于 ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal 中所有大写字母的半径之和F}$$end{document} 涵盖 ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal F}$$end{document} 前提是半径之和小于 π/2。这是Goodman和Goodman所谓的圆覆盖定理的球面类比，也是Jiang和作者证明的Fejes Tóth区域猜想的强化。
A cap of spherical radius α on a unit d-sphere S is the set of points within spherical distance α from a given point on the sphere. Let ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal F}$$end{document} be a finite set of caps lying on S. We prove that if no hyperplane through the center of S divides ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal F}$$end{document} into two non-empty subsets without intersecting any cap in ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal F}$$end{document}, then there is a cap of radius equal to the sum of radii of all caps in ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal F}$$end{document} covering all caps of ℱdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal F}$$end{document} provided that the sum of radii is less than π/2. This is the spherical analog of the so-called Circle Covering Theorem by Goodman and Goodman and the strengthening of Fejes Tóth’s zone conjecture proved by Jiang and the author.