Hadamard Semidifferential, Oriented Distance Function, and some Applications

Abstract

<p style='text-indent:20px;'>The <i>Hadamard semidifferential calculus</i> preserves all the operations of the classical differential calculus including the chain rule for a large family of non-differentiable functions including the continuous convex functions. It naturally extends from the <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional Euclidean space <inline-formula><tex-math id="M2">\begin{document}$ \operatorname{\mathbb R}^n $\end{document}</tex-math></inline-formula> to subsets of topological vector spaces. This includes most function spaces used in <i>Optimization</i> and the <i>Calculus of Variations</i>, the metric groups used in <i>Shape and Topological Optimization</i>, and functions defined on submanifolds.</p><p style='text-indent:20px;'>Certain set-parametrized functions such as the <i>characteristic function</i> <inline-formula><tex-math id="M3">\begin{document}$ \chi_A $\end{document}</tex-math></inline-formula>of a set <inline-formula><tex-math id="M4">\begin{document}$ A $\end{document}</tex-math></inline-formula>, the <i>distance function</i> <inline-formula><tex-math id="M5">\begin{document}$ d_A $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M6">\begin{document}$ A $\end{document}</tex-math></inline-formula>, and the <i>oriented (signed) distance function</i> <inline-formula><tex-math id="M7">\begin{document}$ b_A = d_A-d_{ \operatorname{\mathbb R}^n\backslash A} $\end{document}</tex-math></inline-formula> can be used to identify a space of subsets of <inline-formula><tex-math id="M8">\begin{document}$ \operatorname{\mathbb R}^n $\end{document}</tex-math></inline-formula> with a metric space of set-parametrized functions. Many geometrical properties of domains (convexity, outward unit normal, curvatures, tangent space, smoothness of boundaries) can be expressed in terms of the analytical properties of <inline-formula><tex-math id="M9">\begin{document}$ b_A $\end{document}</tex-math></inline-formula> and a simple intrinsic differential calculus is available for functions defined on hypersurfaces without appealing to local bases or Christoffel symbols.</p><p style='text-indent:20px;'>The object of this paper is to extend the use of the Hadamard semidifferential and of the oriented distance function from finite to infinite dimensional spaces with some selected illustrative applications from shapes and geometries, plasma physics, and optimization.</p>