Abstract: | Let ‖·‖ be the Euclidean norm on R n and γn the (standard) Gaussian measure on R n with density (2π)−n/2e. It is proved that there is a numerical constant c>0 with the following property: if K is an arbitrary convex body in R n with γn(K)≥1/2, then to each sequence u1,…,um∈ R n with ‖u1‖,…,‖um‖≤c there correspond signs ε1,…,εm=±1 such that ∑mi=1εiui∈K. This improves the well-known result obtained by Spencer [Trans. Amer. Math. Soc. 289 , 679–705 (1985)] for the n-dimensional cube. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 351–360, 1998 |