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New metric properties for prox-regular sets
- Source :
- Mathematical Programming
- Publication Year :
- 2020
- Publisher :
- HAL CCSD, 2020.
-
Abstract
- In this paper, we present diverse new metric properties that prox-regular sets shared with convex ones. At the heart of our work lie the Legendre-Fenchel transform and complements of balls. First, we show that a connected prox-regular set is completely determined by the Legendre-Fenchel transform of a suitable perturbation of its indicator function. Then, we prove that such a function is also the right tool to extend, to the context of prox-regular sets, the famous connection between the distance function and the support function of a convex set. On the other hand, given a prox-regular set, we examine the intersection of complements of open balls containing the set. We establish that the distance of a point to a prox-regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point. This is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes. To the best of our knowledge, these results are new in the literature and show that the class of prox-regular sets have good properties known in convex analysis. Mathematics Subject Classification (2010) 49J52 · 49J53
- Subjects :
- Convex analysis
021103 operations research
General Mathematics
Numerical analysis
0211 other engineering and technologies
Regular polygon
Convex set
Mathematics::Optimization and Control
010103 numerical & computational mathematics
02 engineering and technology
Support function
01 natural sciences
Combinatorics
Indicator function
Hyperplane
Metric (mathematics)
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
0101 mathematics
[MATH]Mathematics [math]
Software
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Mathematical Programming
- Accession number :
- edsair.doi.dedup.....026e73705f534633fdd7cd51d5f147a9