On Minimal Absorption Index for an n-Dimensional Simplex

Let and let be the unit cube . For a nondegenerate simplex , by denote the homothetic copy of with center of homothety in the center of gravity of and ratio of homothety Put We call an absorption index of simplex . In the present paper we give new estimates for minimal absorption index of the simplex contained in , i.e., for the number 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 . Always . If there exist an Hadamard matrix of order , then . The best known general upper estimate have the form . There exist constant not depending on such that, for any simplex of maximum volume, inequalities take place. It motivates the making use of maximum volume simplices in upper estimates of . The set of vertices of such a simplex can be consructed with application of maximum -determinant of order or maximum -determinant of order . In the paper we compute absorption indices of maximum volume simplices in constructed from known maximum -determinants via special procedure. For some , this approach makes it possible to lower theoretical upper bounds of . Also we give best known upper estimates of for n <= 118.