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Hadamard Semidifferential, Oriented Distance Function, and some Applications
Département de mathématiques et de statistique and Centre de recherches mathématiques, Université de Montréal, CP 6128, succ. Centreville, Montréal (Qc), Canada H3C 3J7 
The Hadamard semidifferential calculus preserves all the operations of the classical differential calculus including the chain rule for a large family of nondifferentiable functions including the continuous convex functions. It naturally extends from the $ n $dimensional Euclidean space $ \operatorname{\mathbb R}^n $ to subsets of topological vector spaces. This includes most function spaces used in Optimization and the Calculus of Variations, the metric groups used in Shape and Topological Optimization, and functions defined on submanifolds.
Certain setparametrized functions such as the characteristic function $ \chi_A $of a set $ A $, the distance function $ d_A $ to $ A $, and the oriented (signed) distance function $ b_A = d_Ad_{ \operatorname{\mathbb R}^n\backslash A} $ can be used to identify a space of subsets of $ \operatorname{\mathbb R}^n $ with a metric space of setparametrized 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 $ b_A $ and a simple intrinsic differential calculus is available for functions defined on hypersurfaces without appealing to local bases or Christoffel symbols.
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.
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