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Shape analysis via oriented distance functions. (English) Zbl 0814.49032
This paper contains further details of and enhancements on results announced in Lect. Notes Pure Appl. Math. 163, 105-139 (1994; Zbl 0807.49035). Constructions introduced in the work induce topologies on subsets of a fixed domain in $$\mathbb{R}^ n$$ using set parametrized functions. The role of the family of oriented distance functions is studied and it seems that they introduce topologies that retain the classical geometric properties associated with sets: convexity, exterior normals, mean curvature, $$C^ k$$ boundaries, etc.

##### MSC:
 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 28A75 Length, area, volume, other geometric measure theory 26B15 Integration of real functions of several variables: length, area, volume
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