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On the typical structure of compact sets. (English) Zbl 0981.46018

Some properties of typical compact sets in a Banach space are investigated. The set in a metric space is called nowhere dense if its closure has empty interior. A countable union of nowhere dense sets is called to be of the first Baire category. The subset of a complete metric space is said to be residual if its complement is of the first Baire category. If the set of all elements satisfying some property is residual, then this property is called typical or generic.
Let \(E\) be a strictly convex separable Banach space with the dimension at least 2. A compact subset \(K\) of \(E\) is said to have hispit structure if the nearest point mapping \(p_K: E\to 2^K\) is not single-valued on a dense subset of \(E\), that is the set \(A(K)= \{z\in E: \{x\in K:\|x-z\|= d(z,K)\}\) contains at least two points} is dense in \(E\).
The following is proved: If \(K\) is a union of disjoint compact sets \(K_1\) and \(K_2\) and \(K\) has hispit structure, then \(K_1\) and \(K_2\) have also hispit structure. For a typical (in the sense of Baire category) compact set \(K\) and arbitrary \(x\in E\) the set of all \(r>0\) such that the intersection \(K\cap B(x,r)\neq\emptyset\) and has no hispit structure is of Jordan measure zero.

MSC:

46B20 Geometry and structure of normed linear spaces
54E52 Baire category, Baire spaces
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