On Milman's inequality and random subspaces which escape through a mesh in ℝ n

Abstract

Let S be a subset in the Euclidean space ℝ n and 1 <- k < n . We find sufficient conditions which guarantee the existence and even with probability close to 1, of k -codimensional subspaces which miss S . As a consequence we derive a sharp form of Milman's inequality and discuss some applications to Banach spaces.