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Russia Yekaterinburg
Section Mathematics
Title On estimation of Hausdorff deviation of convex polygons in $\mathbb{R}^2$ from their differences with disks
Author(-s) Ushakov V.N.ab, Pershakov M.V.ab
Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb
Abstract We study a problem concerning the estimation of the Hausdorff deviation of convex polygons in $\mathbb R^2$ 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 $\mathbb R^n$ 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 $\mathbb R^n$ in their geometric difference with closed balls in $\mathbb R^n$ 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 $\mathbb R^n$ and the sets approximating them. The paper considers a specific convex heptagon in $\mathbb R^2$. To study the geometry of this heptagon, we introduce the concept of a wedge in $\mathbb R^2$. 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 $\mathbb R^2$ of sufficiently small radius.
Keywords convex polygon in $\mathbb{R}^2$, Hausdorff deviation, wedge, cone, circle, geometric difference of sets
UDC 514.712.2
MSC 52A10, 28A75
DOI 10.35634/vm200404
Received 6 August 2020
Language Russian
Citation Ushakov V.N., Pershakov M.V. On estimation of Hausdorff deviation of convex polygons in $\mathbb{R}^2$ from their differences with disks, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 4, pp. 585-603.
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