Abstract
The object of this paper is twofold. We first present constructions which induce topologies on subsets of a fixed domain or hold-all D in ℝN by using set parametrized functions in an appropriate function space. Secondly we study the role of the family of oriented distance functions (also known as algebraic or signed distance functions) in the analysis of shape optimization problems. They play an important role in the introduction of topologies which retain the classical geometric properties associated with sets: convexity, exterior normals, mean curvature, C k boundaries, etc.
The two authors also acknowledge the support of the Institute for Mathematics and its Applications while attending the IMA Workshop on Control and Optimal Design of Distributed Parameter Systems (November 9–13, 1992).
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Delfour, M.C., Zolésio, JP. (1995). Oriented Distance Functions in Shape Analysis and Optimization. In: Lagnese, J.E., Russell, D.L., White, L.W. (eds) Control and Optimal Design of Distributed Parameter Systems. The IMA Volumes in Mathematics and its Applications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8460-1_3
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