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.