On Characterizations of Submanifolds via Smoothness of the Distance Function in Hilbert Spaces

  • David Salas
  • Lionel ThibaultEmail author


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.


Submanifolds Distance function Metric projection Local uniform continuity Diffeomorphism Prox-regular set Hilbert space 

Mathematics Subject Classification

53B25 49J50 41A65 58C20 46C05 



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Authors and Affiliations

  1. 1.Instituto de Ciencias de la IngenieríaUniversidad de O’HigginsRancaguaChile
  2. 2.CNRS, Institut Montpelliérain Alexander GrothendieckUniversité de MontpellierMontpellierFrance

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