ABSTRACT. We obtain necessary and sufficient conditions for the subdifferentiability and superdifferentiability (in the Demlyanov-Rubinov sense) of the distance in an arbitrary norm from a point to a set for the finitedimensional case. The geometric structure of the subdi_fferential and the superdifferential is described.KEY WORDS: subdifferential, superdifferential, distance function.The distance from a point ~o a 8e~ (the distance function in the sequel) is usually defined as p~(x) = minn(x-y), yen where fl is a closed set in IR p and n(x) is a function satisfying the axioms of norm, and is often used both as a tool and as the object of study (see, for example, [1][2][3][4][5][6][7]). Knowledge of the differential properties of the distance function is essential in estimating and approximating sets and multivalued mappings.The main goal of this paper is to obtain necessary and sufficient conditions for the subdifferentiability and superdifferentiability (in the Dem'yanov-Rubinov sense) of a distance function at a given point x E ]Rp. These are the cases in which the distance function is differentiable at x in all directions and there exist convex compact sets 0~p (x) or Opn(x) such that the derivative of the distance function in any direction 9 can be calculated by either of the formulas rain (w, g).
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