## 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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## References

Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003)

Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys.

**79**, 12–49 (1988)Reiner, T., Mückl, G., Dachsbacher, C.: Interactive modeling of implicit surfaces using a direct visualization approach with signed distance functions. Comput. Graph.

**35**, 596–603 (2011)Armitage, D.H., Kuran, Ü.: The convexity of a domain and the superharmonicity of the signed distance function. Proc. Am. Math. Soc.

**93**, 598–600 (1985)Iwata, S., Nakatsukasa, Y., Takeda, A.: Computing the signed distances between overlapping ellipsoids. SIAM J. Optim.

**25**, 2359–2384 (2015)Delfour, M.C., Zolésio, J.P.: Shape analysis via oriented distance functions. J. Funct. Anal.

**123**, 129–201 (1994)Delfour, M.C., Zolésio, J.P.: Oriented distance function and its evolution equation for initial sets with thin boundary. SIAM J. Control Optim.

**42**, 2286–2304 (2004)Delfour, M.C., Zolésio, J.P.: Oriented Distance Functions in Shape Analysis and Optimization, Control and Optimal Design of Distributed Parameter Systems, pp. 39–72. Springer, New York (1995)

Delfour, M.C., Zolésio, J.P.: Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization, 2nd edn. SIAM, Philadelphia (2001)

Liu, C.G., Ng, K.F., Yang, W.H.: Merit functions in vector optimization. Math. Program. Ser. A.

**119**, 215–237 (2009)Chen, J., Ansari, Q., Yao, J.C.: Characterizations of set order relations and constrained set optimization problems via oriented distance function. Optimization

**66**, 1741–1754 (2017)Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim.

**42**, 1071–1086 (2003)Fuhrmann, A., Sobottka, G., Gross, C.: Abstract distance fields for rapid collision detection in physically based modeling. In: Proceedings of international conference graphicon (2003)

Boczko, E., DiLullo, A., Young, T.: Signed distance functions: a new tool in binary classification. arXiv:CS.LG/0511105 (2005)

Abbasbandy, S., Asady, B.: Ranking of fuzzy numbers by sign distance. Inf. Sci.

**176**, 2405–2416 (2006)Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1999)

Altangerel, L., Wanka, G., Wilfer, O.: An oriented distance function application to gap functions for vector variational inequalities. In: Optimization, simulation, and control, vol. 76. Springer, New York, pp. 17–34 (2013)

Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

Rockafellar, R.T., Wets, J.B.: Variational Analysis. Springer, Berlin (1998)

Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer, New York (2006)

Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)

Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)

Hiriart-Urruty, J.B.: New concepts in nondifferentiable programming. Bull. Soc. Math. France

**60**, 57–85 (1979)Hiriart-Urruty, J.B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res.

**4**, 79–97 (1979)Bot, R.I., Kassay, G., Wanka, G.: Duality for almost convex optimization problems via the perturbation approach. J. Glob. Optim.

**42**, 385–399 (2008)Moffat, S.M., Moursi, W.M., Wang, X.: Nearly convex sets: fine properties and domains or ranges of subdifferentials of convex functions. Math. Program. Ser. A.

**160**, 193–223 (2016)Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)

Brezis, H., Haraux, A.: Image d’une somme d’operateurs monotones et applications. Israel J. Math.

**23**, 165–186 (1976)Simons, S.: From Hahn-Banach to Monotonicity. Springer, New York (2008)

Borwein, J.M., Giles, J.R.: The proximal normal formula in Banach space. Trans. Am. Math. Soc.

**302**, 371–381 (1987)Preiss, D., Zajćek, L.: Fréchet differentiation of convex functions in a Banach space with a separable dual. Proc. Am. Math. Soc.

**91**, 202–204 (1994)Mangasarian, O.L.: Polyhedral boundary projection. SIAM J. Optim.

**9**, 1128–1134 (1999)Boyd, S., Vandenberghe, L.: Convex Optimization, Solutions Manual. Cambridge University Press, Cambridge (2006)

Hager, W.W., Zhang, H.C.: Projector onto a polyhedron that exploits sparsity. SIAM J. Optim.

**29**, 1773–1798 (2016)Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

Bauschke, H.H., Moffat, S.M., Wang, X.: Near equality, near convexity, sums of maximally monotone operators, and averages of firmly nonexpansive mappings. Math. Program. Ser. B.

**139**, 55–70 (2013)Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

## 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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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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DOI: https://doi.org/10.1007/s10957-018-1414-2

### Keywords

- Boundary projection
- Fenchel conjugate
- Maximally monotone operator
- Nearly convex sets
- Not convex subdifferential domain
- Paramonotone operator
- Signed distance function
- Skeleton of a convex set
- Subdifferential