On one addition to evaluation by L.\,S.~Pontryagin of the geometric difference of sets in a plane

V. N. Ushakov ; A. A. Ershov ; M. V. Pershakov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 54 (2019), pp. 63-73

Résumé

In this paper, two generalizations of convex sets on the plane are considered. The first generalization is the concept of the $\alpha$-sets. These sets allow for the existence of several projections onto them from an arbitrary point on the plane. However, these projections should be visible from this point at an angle not exceeding $\alpha$. The second generalization is related to the definition of a convex set according to which the segment connecting the two points of the convex set is also inside it. We consider central symmetric sets for which this statement holds only for two points lying on the opposite sides of some given line. For these two types of nonconvex sets, the problem of finding the maximum area subset is considered. The solution to this problem can be useful for finding suboptimal solutions to optimization problems and, in particular, linear programming. A generalization of the Pontryagin estimate for the geometric difference of an $\alpha$-set and a ball is proved. In addition, as a corollary, the statement that the $\alpha$-set in the plane necessarily contains a nonzero point with integer coordinates if its area exceeds a certain critical value is given. This corollary is one of generalizations of the Minkowski theorem for nonconvex sets.

Détail

Mots-clés : $\alpha$-set
Keywords: Minkowski theorem, nonconvex set, convex subset, geometric difference.