Spherical Shell of the Boundary of a Compact Set with a Minimum Cross-Sectional Area Formed by a Two-Dimensional Plane

Abstract

For a given compact set, the finite-dimensional problem of constructing a spherical shell of its boundary such that the shell cross section formed by a two-dimensional plane passing through its center has a minimum area is considered. It is proved that the problem has a solution, and a criterion is found under which the solution set is bounded. The objective function of the given optimization problem is shown to be convex, and a formula for its subdifferential is derived. A criterion for solving the problem is obtained, which is used to establish some properties of the solution and to find conditions for solution uniqueness. In the two-dimensional case when the compact set is a convex body, it is proved that the solution sets of the given problem and the asphericity problem for this body intersect at a single point that is the solution of the problem of finding a least-thickness spherical shell of the boundary of the given body.