Abstract
Given a closed subset $S$ of a Banach space $X$, we study the minimal time function to reach a point $x$ of $X$ starting from $S$. We relate this problem to the study of the geodesic distance from $x$ to $S$ associated with a Riemannian metric or a Finsler metric. It appears that both cases can be treated simultaneously and yield solutions to a Hamilton--Jacobi equation. We detect simple links between the general framework we propose and multivalued semigroups. New concepts of infinitesimal generators of such semigroups are considered.
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