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On Milman’s inequality and random subspaces which escape through a mesh in \({\mathbb{R}}^ n\). (English) Zbl 0651.46021
Geometric aspects of functional analysis, Isr. Semin. 1986-87, Lect. Notes Math. 1317, 84-106 (1988).
[For the entire collection see Zbl 0638.00019.]
This sharp and not easily readable paper contains results of the following type: find sufficient conditions which guarantee the existence (and even, with probability close to 1), of k-dimensional subspaces which miss a subset S of the Euclidean space \({\mathbb{R}}^ n\) with \(1\leq k<n\). Also, the following other facts are studied: the possibility to hit S and miss a “piece” of S; the distance between a randomly chosen k- codimensional subspace and convex sets situated at various locations in \({\mathbb{R}}^ n\). Results related to the problems considered here are the following: isoperimetric inequalities on the sphere; quantitative versions of Dvoretzki theorem for subspaces of quotients of finite dimensional Banach spaces; sharp versions of Milman’s inequality.
Reviewer: P.Papini

46B20 Geometry and structure of normed linear spaces