On estimation of Hausdorff deviation of convex polygons in R2 from their differences with disks

Abstract

We study a problem concerning the estimation of the Hausdorff deviation of convex polygons in R2 from their geometric difference with circles of sufficiently small radius. Problems with such a subject, in which not only convex polygons but also convex compacts in the Euclidean space Rn are considered, arise in various fields of mathematics and, in particular, in the theory of differential games, control theory, convex analysis. Estimates of Hausdorff deviations of convex compact sets in Rn in their geometric difference with closed balls in Rn are presented in the works of L.S. Pontryagin, his staff and colleagues. These estimates are very important in deriving an estimate for the mismatch of the alternating Pontryagin's integral in linear differential games of pursuit and alternating sums. Similar estimates turn out to be useful in deriving an estimate for the mismatch of the attainability sets of nonlinear control systems in Rn and the sets approximating them. The paper considers a specific convex heptagon in R2. To study the geometry of this heptagon, we introduce the concept of a wedge in R2. On the basis of this notion, we obtain an upper bound for the Hausdorff deviation of a heptagon from its geometric difference with the disc in R2 of sufficiently small radius. © 2020 Udmurt State University. All rights reserved.