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On characterizations of submanifolds via smoothness of the distance function in Hilbert spaces. (English) Zbl 1433.46016

Summary: The property of continuous differentiability with Lipschitz derivative of the square distance function is known to be a characterization of prox-regular sets. We show in this paper that the property of higher-order continuous differentiability with locally uniformly continuous last derivative of the square distance function near a point of a set characterizes, in Hilbert spaces, that the set is a submanifold with the same differentiability property near the point.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
49J50 Fréchet and Gateaux differentiability in optimization
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