Individual approximative properties of sets and approximative compactness

A point $x\in X$ is a point of approximative compactness for a set $\emptyset \ne M\subset X$ if each minimizing sequence of points from~$M$ for~$x$ contains a~subsequence converging to some point from~$M$. E.\,V.~Oshman established that each convex set of existence is approximatively compact if and only if so is each proximinal hyperplane. Given a~set~$M$, we introduce the set of $M$-acting points (the range of the normalized metric projection onto the set~$M$), which is an individual approximation characteristics of the set~$M$. In terms of this characteristics, we give conditions on a~space~$X$ which guarantee that a~given set~$M$ is approximatively (strongly or weakly) compact.

УДК 517.982.256

Keywords: individual approximation, approximatively compact set, approximatively weakly compact set, neighborly $P$-convex set, Day--Oshman space, stability of the distance minimization problem.