On the guaranteed estimates of the area of convex subsets of compacts on a plane

Vladimir N. Ushakov, Alexander A. Ershov

IMM UB RAS

Abstract: The paper considers the problem of constructing a convex subset of the largest area in a nonconvex compact on the plane, as well as the problem of constructing a convex subset from which the Hausdorff deviation of the compact is minimal. Since, in the general case, the exact solution of these problems is impossible, the geometric difference between the convex hull of a compact and a circle of a 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 considered the problem of constructing convex subsets from an $\alpha$-set and a set with a finite Mordell concavity coefficient.

Keywords: convex set, geometric difference, $\alpha$-set, Mordell concavity ratio, figure area, Hausdorff deviation.

UDC: 517.977
BBK: 22.18

Received: 06.11.2020
Revised: 08.12.2020
Accepted: 10.12.2020