Variational analysis for a class of minimal time functions in Hilbert spaces

This paper considers the parameterized infinite dimensional optimization problem \[ \text{minimize}\quad\bigl\{t\geq 0:\;S\cap\{x+tF\}\not= \emptyset\bigr\}, \] where $S$ is a nonempty closed subset of a Hilbert space $H$ and $F\subseteq H$ is closed convex satisfying $0\in\iint F$. The optimal value $\tsf(x)$ depends on the parameter $x\in H$, and the (possibly empty) set $S\cap (x+T(x)F)$ of optimal solutions is the ``$F$-projection'' of $x$ into $S$. We first compute proximal and Fr\'echet subgradients of $\tsf(\cdot)$ in terms of normal vectors to level sets, and secondly, in terms of the $F$-projection. Sufficient conditions are also obtained for the differentiability and semiconvexity of $\tsf(\cdot)$, results which extend the known case when $F$ is the unit ball.