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Transactions of the American Mathematical Society

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Differentiability of the metric projection in Hilbert space


Authors: Simon Fitzpatrick and R. R. Phelps
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
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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 $ \S3$ 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 $ \S5$ 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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0645326-5
Keywords: Gateaux derivative, Fréchet derivative, metric projection, nearest-point map, convex body, Hilbert space, Minkowski functional, gauge functional
Article copyright: © Copyright 1982 American Mathematical Society

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