Differentiability of the metric projection in Hilbert space
Simon Fitzpatrick, R. R. Phelps · Transactions of the American Mathematical Society · 1982
A study is made of differentiability of the metric projection P P onto a closed convex subset K K of a Hilbert space H H . When K 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 P . For instance, combining results in § 3 \S 3 with earlier work of R. D. Holmes shows that K K has a C 2 {C^2} boundary if and only if P P is C 1 {C^1} in H ∖ K H\backslash K and its derivative P ′ P’ has a certain invertibility property at each point. An example in § 5 \S 5 shows that if the C 2 {C^2} condition is relaxed even slightly then P P can be nondifferentiable (Fréchet) in H ∖ K H\backslash K