Differentiability of the metric projection in Hilbert space
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- by Simon Fitzpatrick and R. R. Phelps PDF
- Trans. Amer. Math. Soc. 270 (1982), 483-501 Request permission
Abstract:
A study is made of differentiability of the metric projection $P$ onto a closed convex subset $K$ of a Hilbert space $H$. When $K$ has nonempty interior, the Gateaux or Fréchet smoothness of its boundary can be related with some precision to Gateaux or Fréchet differentiability properties of $P$. For instance, combining results in $\S 3$ with earlier work of R. D. Holmes shows that $K$ has a ${C^2}$ boundary if and only if $P$ is ${C^1}$ in $H\backslash K$ and its derivative $P’$ has a certain invertibility property at each point. An example in $\S 5$ shows that if the ${C^2}$ condition is relaxed even slightly then $P$ can be nondifferentiable (Fréchet) in $H\backslash K$.References
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Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 270 (1982), 483-501
- MSC: Primary 41A65; Secondary 41A50, 46C99, 58B20, 58C20
- DOI: https://doi.org/10.1090/S0002-9947-1982-0645326-5
- MathSciNet review: 645326