Hadamard Semidifferential, Oriented Distance Function, and some Applications

<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 $\end{document}</tex-math></inline-formula>-dimensional Euclidean space id="M2">\begin{document}$ \operatorname{\mathbb R}^n $\end{document}</tex-math></inline-formula> to subsets topological vector spaces. This includes most function spaces used in <i>Optimization</i> and <i>Calculus Variations</i>, metric groups <i>Shape Topological Optimization</i>, defined on submanifolds.</p><p style='text-indent:20px;'>Certain set-parametrized such as <i>characteristic function</i> id="M3">\begin{document}$ \chi_A $\end{document}</tex-math></inline-formula>of set id="M4">\begin{document}$ A $\end{document}</tex-math></inline-formula>, <i>distance id="M5">\begin{document}$ d_A id="M6">\begin{document}$ <i>oriented (signed) distance id="M7">\begin{document}$ b_A = d_A-d_{ R}^n\backslash A} can be identify id="M8">\begin{document}$ with Many geometrical properties domains (convexity, outward unit normal, curvatures, tangent space, smoothness boundaries) expressed terms analytical id="M9">\begin{document}$ simple intrinsic is available hypersurfaces without appealing local bases or Christoffel symbols.</p><p object this paper extend use Hadamard oriented finite infinite dimensional some selected illustrative applications shapes geometries, plasma physics, optimization.</p>