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Variational Analysis on the Signed Distance Functions

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Abstract

The signed distance function (or oriented distance function) of a set in a metric space determines the distance of a given point from the boundary of the set, with the sign determined by whether the point is in the set or in its complement. The knowledge of signed distance functions is a very valuable information in various fields of applied mathematics such as collision detection, binary classification, shape analysis, fuzzy numbers ranking and level set methods. One distinguished feature of the signed distance function is that it reflects the geometric structure of the set much better than the distance function does. We explore many interesting analytical properties of signed distance functions and use them to construct convex functions with not convex subdifferential domains. Several examples are presented to illustrate most of these fine properties.

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Acknowledgements

We are very grateful to two anonymous referees and Associate Editor C. Zalinescu whose insightful comments helped to considerably improve an earlier version of this paper. Honglin Luo was supported by NSFC (Nos. 11601050, 11431004) and CQNSF (Nos. KJ1600316, cstc2016jcyjA0116). Xianfu Wang was partially supported by the Natural Sciences and Engineering Research Council of Canada.

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Correspondence to Xianfu Wang.

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Communicated by Constantin Zalinescu.

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Luo, H., Wang, X. & Lukens, B. Variational Analysis on the Signed Distance Functions. J Optim Theory Appl 180, 751–774 (2019). https://doi.org/10.1007/s10957-018-1414-2

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