Geodesic
On one addition to evaluation by L.\,S.~Pontryagin of the geometric difference of sets in a plane
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 54 (2019), pp. 63-73.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Mots-clés : $\alpha$-set
Keywords: Minkowski theorem, nonconvex set, convex subset, geometric difference. 
@article{IIMI_2019_54_a5,
     author = {V. N. Ushakov and A. A. Ershov and M. V. Pershakov},
     title = {On one addition to evaluation by {L.\,S.~Pontryagin} of the geometric difference of sets in a plane},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {63--73},
     publisher = {mathdoc},
     volume = {54},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2019_54_a5/}
}
TY  - JOUR
AU  - V. N. Ushakov
AU  - A. A. Ershov
AU  - M. V. Pershakov
TI  - On one addition to evaluation by L.\,S.~Pontryagin of the geometric difference of sets in a plane
JO  - Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
PY  - 2019
SP  - 63
EP  - 73
VL  - 54
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IIMI_2019_54_a5/
LA  - ru
ID  - IIMI_2019_54_a5
ER  - 
%0 Journal Article
%A V. N. Ushakov
%A A. A. Ershov
%A M. V. Pershakov
%T On one addition to evaluation by L.\,S.~Pontryagin of the geometric difference of sets in a plane
%J Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
%D 2019
%P 63-73
%V 54
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IIMI_2019_54_a5/
%G ru
%F IIMI_2019_54_a5
V. N. Ushakov; A. A. Ershov; M. V. Pershakov. On one addition to evaluation by L.\,S.~Pontryagin of the geometric difference of sets in a plane. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 54 (2019), pp. 63-73. http://geodesic.mathdoc.fr/item/IIMI_2019_54_a5/

[1] Uspenskii A. A., Ushakov V. N., Fomin A. N., $\alpha$-sets and their properties, Deposited in VINITI 02.04.2004, No 543-B2004, IMM UB RAS, Yekaterinburg, 2004, 62 pp. (in Russian)

[2] Ivanov G. E., Weak convex sets and functions: theory and applications, Fizmatlit, M., 2006

[3] Zelinskii Yu.B., Convexity. Selected chapters, Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, 2012

[4] Ngai H. V., Penot J.-P., “Paraconvex functions and paraconvex sets”, Studia Mathematica, 184:1 (2008), 1–29 | DOI | MR | Zbl

[5] Semenov P. V., “Functionally paraconvex sets”, Mathematical Notes, 54:6 (1993), 1236–1240 | DOI | MR | Zbl

[6] Ershov A. A., Pershakov M. V., “On matching up the alpha-sets with other generalizations of convex sets”, VI Information school of a young scientist, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 2018, 143–150 (in Russian) | DOI

[7] Ivanov G. E., “Weakly convex sets and their properties”, Mathematical Notes, 79:1–2 (2006), 55–78 | DOI | DOI | MR | Zbl

[8] Ivanov G. E., Polovinkin E. S., “Second-order convergence of an algorithm calculating the value of linear differential games”, Doklady Mathematics, 51:1 (1995), 29–32 | Zbl

[9] Ivanov G. E., Golubev M. O., “Strong and weak convexity in nonlinear differential games”, IFAC-PapersOnline, 51:32 (2018), 13–18 | DOI

[10] Pontrjagin L. S., “Linear differential games of pursuit”, Mathematics of the USSR-Sbornik, 40:3 (1981), 285–303 | DOI | Zbl

[11] Gruber P. M., Lekkerkerker C. G., Geometry of numbers, North-Holland, 1987 | MR | Zbl

[12] Mahler K. Ein Übertragunsprinzip für konvexe Kǒrper, Časopis Pěst. Mat. Fys., 68 (1939), 93–102 | MR

[13] Sawyer D. B., “The lattice determinants of asymmetric convex regions”, J. London Math. Soc., 29 (1954), 251–254 | DOI | MR | Zbl

[14] Polovinkin E. S., Balashov M. V., Elements of convex and strongly convex analysis, Fizmatlit, M., 2007

[15] Starr R. M., “Quasi-equilibria in markets with non-convex preferences”, Econometrica, 37:1 (1969), 25–38 | DOI | Zbl

[16] Garkavi A. L., “On the Chebyshev center and convex hull of a set”, Uspekhi Mat. Nauk, 19:6 (1964), 139–145 (in Russian) | Zbl

[17] Ushakov V. N., Ershov A. A., “An estimate of the Hausdorff distance between a set and its convex hull in Euclidean spaces of small dimension”, Trudy Instituta Matematiki i Mekhaniki URO RAN, 24, no. 1, 223–235 (in Russian) | DOI