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On Guaranteed Estimates of the Area of Convex Subsets of Compact Sets on the Plane

  • MATHEMATICAL GAME THEORY AND APPLICATIONS
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Abstract

The paper considers the problem of constructing a convex subset of the largest area in a nonconvex compact set on the plane as well as the problem of constructing a convex subset such that the Hausdorff deviation of the compact set from this subset is minimal. Since the exact solution of these problems is impossible in the general case, the geometric difference between the convex hull of the compact set and a circle of certain radius is proposed as an acceptable replacement for the exact solution. A lower bound for the area of this geometric difference and an upper bound for the Hausdorff deviation from it of a given nonconvex compact set are obtained. As examples, we consider the problem of constructing convex subsets in an \(\alpha \)-set and a set with a finite Mordell concavity coefficient.

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ACKNOWLEDGMENTS

The present authors know Leon Aganesovich Petrosyan as one of the most authoritative specialists in the theory of differential games, who stood at the origins of this theory in our country, and as the founder and leader of a large scientific school. We deeply respect his scientific achievement and the energy that he puts into scientific research. We wish him good health and success in science.

Funding

This work was financially supported by the Russian Foundation for Basic Research, project no. 18–01–00221 A.

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Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620108, Russia

    V. N. Ushakov & A. A. Ershov

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  1. V. N. Ushakov
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  2. A. A. Ershov
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Correspondence to V. N. Ushakov or A. A. Ershov.

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Translated by V. Potapchouck

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Ushakov, V.N., Ershov, A.A. On Guaranteed Estimates of the Area of Convex Subsets of Compact Sets on the Plane. Autom Remote Control 82, 1976–1984 (2021). https://doi.org/10.1134/S0005117921110126

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  • DOI: https://doi.org/10.1134/S0005117921110126

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