Differentiability of the metric projection in Hilbert space

Fitzpatrick, Simon; Phelps, R. R.

doi:10.1090/S0002-9947-1982-0645326-5

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 $H\backslash K$.

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