Abstract

Let \(C\) be a convex body and let \(S\) be a nondegenerate simplex in \({{\mathbb{R}}^{n}}\). Denote by \(\tau S\) the image of \(S\) under the homothety with center of homothety in the center of gravity of \(S\) and ratio of homothety \(\tau \). We mean by \(\xi (C;S)\) the minimal \(\tau > 0\) such that \(C\) is a subset of the simplex \(\tau S\). Define \(\alpha (C;S)\) as the minimal \(\tau > 0\) such that \(C\) is contained in a translate of \(\tau S\). Earlier the author has proved the equalities \(\xi (C;S) = (n + 1)\mathop {\max }\limits_{1 \leqslant j \leqslant n + 1} \mathop {\max }\limits_{x \in C} ( - {{\lambda }_{j}}(x)) + 1\) (if \(C { \text{⊄} }S\)), \(\alpha (C;S) = \sum\nolimits_{j = 1}^{n + 1} {\mathop {\max }\limits_{x \in C} ( - {{\lambda }_{j}}(x)) + 1.} \) Here \({{\lambda }_{j}}\) are linear functions called the basic Lagrange polynomials corresponding to \(S\). The numbers \({{\lambda }_{j}}(x), \ldots ,{{\lambda }_{{n + 1}}}(x)\) are the barycentric coordinates of a point \(x \in {{\mathbb{R}}^{n}}\). In his previous papers, the author has investigated these formulae in the case when \(C\) is the \(n\)-dimensional unit cube \({{Q}_{n}} = {{[0,1]}^{n}}\). The present paper is related to the case when \(C\) coincides with the unit Euclidean ball \({{B}_{n}} = \{ x:\left| {\left| x \right|} \right| \leqslant 1\} ,\) where \(\left| {\left| x \right|} \right| = \mathop {\left( {\sum\nolimits_{i = 1}^n {x_{i}^{2}} \,} \right)}\nolimits^{1/2} .\) We establish various relations for \(\xi ({{B}_{n}};S)\) and \(\alpha ({{B}_{n}};S)\), as well as we give their geometric interpretation. For example, if \({{\lambda }_{j}}(x){{l}_{{1j}}}{{x}_{1}} + \ldots + {{l}_{{nj}}}{{x}_{n}} + {{l}_{{n + 1,j}}},\) then \(\alpha ({{B}_{n}};S) = \sum\nolimits_{j = 1}^{n + 1} {{{{\left( {\sum\nolimits_{i = 1}^n {l_{{ij}}^{2}} } \right)}}^{{{\text{1}}{\text{/}}{\text{2}}}}}} \). The minimal possible value of each characteristic \(\xi ({{B}_{n}};S)\) and \(\alpha ({{B}_{n}};S)\) for \(S \subset {{B}_{n}}\) is equal to \(n\). This value corresponds to a regular simplex inscribed into \({{B}_{n}}\). Also we compare our results with those obtained in the case \(C = {{Q}_{n}}\).

中文翻译：

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关于$$ {{\ mathbb {R}} ^ {n}} $$中的单纯形和球形的一些问题

摘要

令\（C \）为凸体，令\（S \）为\（{{\ mathbb {R}} ^ {n}} \）中的非简并单纯形。用\（\ tau S \）表示同质下的\（S \）图像，其中同质中心在\（S \）的重心和同质比\（\ tau \）中。我们的意思是\（\ xi（C; S）\）最小\（\ tau> 0 \），使得\（C \）是单纯形\（\ tau S \）的子集。将\（\ alpha（C; S）\）定义为最小\（\ tau> 0 \），以使\（C \）包含在\（\ tau S \）的转换中。早前作者证明了等式\（\ xi（C; S）=（n + 1）\ mathop {\ max} \ limits_ {1 \ leqslant j \ leqslant n + 1} \ mathop {\ max} \ limits_ { x \ in C}（-{{\ lambda __ {j}}（x））+ 1 \）（如果\（C {\ text {⊄}} S \）），\（\ alpha（C; S ）= \ sum \ nolimits_ {j = 1} ^ {n + 1} {\ mathop {\ max} \ limits_ {x \ in C}（-{{\ lambda __ {j}}（x））+ 1 。} \）这里的\（{{\ lambda} _ {j}} \）是称为\（S \）的基本拉格朗日多项式的线性函数。数字\（{{\ lambda __j}}（x），\ ldots，{{\ lambda} _ {{n + 1}}}（x）\）是点\（x的重心坐标\ in {{\ mathbb {R}} ^ {n}} \）中。在以前的论文中，作者研究了以下公式\（C \）是\（n \）维单位立方\（{{Q} _ {n}} = {{[0,1]} ^ {n}} \）。本文涉及\（C \）与单位欧几里得球\（{{B} _ {n}} = \ {x：\ left | {\ left | x \ right |} \ right | \ leqslant 1 \}，\）其中\（\ left | {\ left | x \ right |} \ right | = \ mathop {\ left（{\ sum \ nolimits_ {i = 1} ^ n {x_ {i } ^ {2}} \，} \ right）} \ nolimits ^ {1/2}。\）我们为\（\ xi（{{B} _ {n}}; S）\）和\建立各种关系（\ alpha（{{B} _ {n}}; S）\），以及我们给出的几何解释。例如，如果\（{{\ lambda} _ {j}}（x）{{l} _ {{1j}}} {{x} _ {1}} + \ ldots + {{l} _ {{ nj}}} {{x} _ {n}} + {{l} _ {{n + 1，j}}}，\）然后\（\ alpha（{{B} _ {n}}; S）= \ sum \ nolimits_ {j = 1} ^ {n + 1} {{{{\ left（{\ sum \ nolimits_ {i = 1} ^ n {l _ {{ij}} ^ {2}}} \ right）}} ^ {{{\ text {1}} {\ text {/}} {\ text {2}}}}}}} \}。各特性的最小可能值\（\ XI（{{B} _ {N}}; S）\）和\（\α（{{B} _ {N}}; S）\）为\（S \ subset {{B} _ {n}} \）等于\（n \）。该值对应于\（{{B} _ {n}} \）内嵌的常规单纯形。另外，我们将结果与在\（C = {{Q} _ {n}} \）情况下获得的结果进行比较。