A Formula for the Superdifferential of the Distance Determined by the Gauge Function to the Complement of a Convex Set

The distance determined by the Minkowski gauge function to the complement of a convex solid body in a finite-dimensional space is considered. The concavity of this distance function on a given convex set is proved, and a formula for its superdifferential at any interior point of this set is obtained...

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Bibliographic Details
Published inMathematical Notes Vol. 106; no. 5-6; pp. 703 - 710
Main Authors Dudov, S. I., Osiptsev, M. A.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.11.2019
Springer Nature B.V
Subjects
Concavity
Mathematics
Mathematics and Statistics
gauge function of a set
distance function
superdifferential
support function
cone of possible directions
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Summary:The distance determined by the Minkowski gauge function to the complement of a convex solid body in a finite-dimensional space is considered. The concavity of this distance function on a given convex set is proved, and a formula for its superdifferential at any interior point of this set is obtained. It is also proved that the distance function under consideration is directionally differentiable at the boundary points of the convex set, and formulas for its directional derivative are obtained.
ISSN:0001-4346
1067-9073
1573-8876
DOI:10.1134/S000143461911004X