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JOURNALS // Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki // Archive

Zh. Vychisl. Mat. Mat. Fiz., 2017 Volume 57, Number 12, Pages 1946–1954 (Mi zvmmf10647)

Inscribed balls and their centers

M. V. Balashov

Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow oblast, Russia

Abstract: A ball of maximal radius inscribed in a convex closed bounded set with a nonempty interior is considered in the class of uniformly convex Banach spaces. It is shown that, under certain conditions, the centers of inscribed balls form a uniformly continuous (as a set function) set-valued mapping in the Hausdorff metric. In a finite-dimensional space of dimension $n$, the set of centers of balls inscribed in polyhedra with a fixed collection of normals satisfies the Lipschitz condition with respect to sets in the Hausdorff metric. A Lipschitz continuous single-valued selector of the set of centers of balls inscribed in such polyhedra can be found by solving $n+1$ linear programming problems.

Key words: inscribed ball, center of an inscribed ball, Hausdorff metric, uniform continuity, uniform convexity, Lipschitz condition, linear programming.

UDC: 519.6

Received: 20.12.2016
Revised: 26.02.2017

DOI: 10.7868/S0044466917120080


 English version:
Computational Mathematics and Mathematical Physics, 2017, 57:12, 1899–1907

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© Steklov Math. Inst. of RAS, 2025