Approximative solar properties of sets and local geometry of the unit sphere

Алимов, А.Р.

Автор: Алимов Алексей Ростиславович
Журнал: Lobachevskii Journal of Mathematics
Том: 12
Год издания: 2023
Издательство: Kazanskii Gosudarstvennyi Universitet/Kazan State University
Местоположение издательства: Russian Federation
Первая страница: 5148.
Последняя страница: 5154
DOI: 10.1134/S1995080223120053
Аннотация: Given a~nonempty subset~$M$ of a~normed space, a~point~$s$ of theunit sphere is said to be $M$-acting if some ball $B(x,r)$ touchesthe set~$M$ by an analogue~$y$ of the point~$s$, i.e., $s=(y-x)/r$.Balayage theorems of geometric approximation theory are obtainedin terms of $M$-acting points. The concepts of $M$-strictly convexand $M$-uniformly convex spaces are introduced, density results forpoints of uniqueness of closed sets are obtained. In particular, wegeneralize one result of S.\,B.~Stechkin, who characterized thespaces~$X$, where the set of points of uniqueness for any subsetof~$X$ is dense in~it. We also show that if $M\subset X$ is a~sunand if any $M$-acting luminosity point is an {\rm LUR}-point of theunit sphere, then $M$ has connected intersections with closedballs. A kind of converse of this result is proved: if a givenset~$M$ has connected intersections with open balls and if any$M$-acting luminosity point is an LUR-point, then $M$ is a~unimodalset, i.e., each local minimum of the distance function to M is aglobal one.
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