Abstract
Let $$n \in \mathbb{N}$$ and let $${{Q}_{n}}$$ be the unit cube $${{[0,1]}^{n}}$$ . For a nondegenerate simplex $$S \subset {{\mathbb{R}}^{n}}$$ , by $$\sigma S$$ denote the homothetic copy of $$S$$ with center of homothety in the center of gravity of $$S$$ and ratio of homothety $$\sigma .$$ Put $$\xi (S) = min{\text{\{ }}\sigma \geqslant 1:{{Q}_{n}} \subset \sigma S{\text{\} }}{\text{.}}$$ We call $$\xi (S)$$ an absorption index of simplex $$S$$ . In the present paper we give new estimates for minimal absorption index of the simplex contained in $${{Q}_{n}}$$ , i.e., for the number $${{\xi }_{n}} = min{\text{\{ }}\xi (S):S \subset {{Q}_{n}}{\text{\} }}.$$ In particular, this value and its analogues have applications in estimates for the norms of interpolation projectors. Previously the first author proved some general estimates of $${{\xi }_{n}}$$ . Always $$n \leqslant {{\xi }_{n}} < n + 1$$ . If there exist an Hadamard matrix of order $$n + 1$$ , then $${{\xi }_{n}} = n$$ . The best known general upper estimate have the form $${{\xi }_{n}} \leqslant \tfrac{{{{n}^{2}} - 3}}{{n - 1}}$$ $$(n > 2)$$ . There exist constant $$c > 0$$ not depending on $$n$$ such that, for any simplex $$S \subset {{Q}_{n}}$$ of maximum volume, inequalities $$c\xi (S) \leqslant {{\xi }_{n}} \leqslant \xi (S)$$ take place. It motivates the making use of maximum volume simplices in upper estimates of $${{\xi }_{n}}$$ . The set of vertices of such a simplex can be consructed with application of maximum $$0/1$$ -determinant of order $$n$$ or maximum $$ - 1/1$$ -determinant of order $$n + 1$$ . In the paper we compute absorption indices of maximum volume simplices in $${{Q}_{n}}$$ constructed from known maximum $$ - 1/1$$ -determinants via special procedure. For some $$n$$ , this approach makes it possible to lower theoretical upper bounds of $${{\xi }_{n}}$$ . Also we give best known upper estimates of $${{\xi }_{n}}$$ for $$n \leqslant 118$$ .
Highlights
by σS denote the homothetic copy of S with center
we give new estimates for the minimal absorption index of the simplex contained in Qn
Previously the first author proved some general estimates of ξn
Summary
Для невырожденного симплекса S ⊂ Rn через σS 0, такая что для любого симплекса S ⊂ Qn, имеющего максимальный объём, выполняются неравенства cξ(S) ≤ ξn ≤ ξ(S). Ю., "О минимальном коэффициенте поглощения для n-мерного симплекса", Моделирование и анализ информационных систем, 25:1 (2018), 140–150. Ю. О минимальном коэффициенте поглощения для n-мерного симплекса
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