Properties of the Distance Function to Strongly and Weakly Convex Sets in a Nonsymmetrical Space

We consider the distance function (DF), given by the caliber (the Minkowski gauge function) of a convex body, from a point to strictly, strongly, and weakly convex sets in an arbitrary Hilbert space. Some properties of the caliber of a strongly convex set and the conditions for obtaining a strict, s...

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Bibliographic Details
Published inRussian mathematics Vol. 64; no. 5; pp. 17 - 30
Main Authors Dudov, S. I., Polovinkin, E. S., Abramova, V. V.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.05.2020
Springer Nature B.V
Subjects
Convexity
Data envelopment analysis
Hilbert space
Mathematics
Mathematics and Statistics
strongly and weakly convex sets and functions
gauge of a set
distance function (DF)
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Summary:We consider the distance function (DF), given by the caliber (the Minkowski gauge function) of a convex body, from a point to strictly, strongly, and weakly convex sets in an arbitrary Hilbert space. Some properties of the caliber of a strongly convex set and the conditions for obtaining a strict, strong, or weak convexity of Lebesgue sets for the distance function are established in accordance with the requirements for the set, the caliber of which specifies the distance function, and the set to which the distance is measured. The corresponding inequalities are obtained that reflect the behavior of the distance function on segments and allow comparing it with strictly, strongly, or weakly convex functions.
ISSN:1066-369X
1934-810X
DOI:10.3103/S1066369X20050035