In this paper, we provide in a general Hilbert space new characterizations of uniform prox-regularity involving outside but sufficiently close points considered sets. We show that the complement prox-regular set is nothing union closed balls with common radius. derive from given equivalent to semiconvexity property its distance function. Various estimates metric projection mapping are also esta...

<p style='text-indent:20px;'>The <i>Hadamard semidifferential calculus</i> preserves all the operations of classical differential calculus including chain rule for a large family non-differentiable functions continuous convex functions. It naturally extends from <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formul...