Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces

E. D. Gluskin

Sbornik. Mathematics, Tome 64 (1989) no. 1, pp. 85-96

Résumé

It is proved that the distribution function for the maximum of the modulus of a set $n$ of jointly Gaussian random variables with given variance and zero mean is minimal if these variables are independent. For $n\leqslant N$ let $$ \alpha_{N,n}=\sup_{x_1,\dots,x_N\in B_2^n}\inf_{z\in S^{n-1}}\sup_{1\leqslant j\leqslant N}|\langle x_j,z\rangle|. $$ As a corollary of the result mentioned, the precise orders of the constants $\alpha_{N,n}$ are computed $\alpha_{N,n}\asymp\min\{1,\sqrt{n^{-1}\log(1+N/n)}\}$, and various improvements of these inequalities are obtained. The estimates are used in particular to construct lacunary analogues of the Rudin–Shapiro trigonometric polynomials. Bibliography: 23 titles.