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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

##### MSC:
 46B20 Geometry and structure of normed linear spaces