Subdifferentiability and superdifferentiability of distance functions

Abstract

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).

Uniform Estimation of a Convex Body by a Fixed-Radius Ball

“…Now we prove the reverse inclusion to (15). Indeed, assuming the contrary, we see that there must exist x * / ∈ C(r ) and x * ∈ C φ (r ).…”

“…The function ρ D (x) is a finite convex function over R p , and the formula for its subdifferential at every x ∈ R p is as follows [15] ∂ρ…”

“…The function ρ (x) is concave on D, and its superdifferential at points x ∈ int D is defined by the formula [15]:…”

Систематизация Задач По Шаровым Оценкам Выпуклого Компакта

“…Если компакт D является выпуклым, то функция ρ D (x) является выпуклой и конечной на R p , а ее субдифференциал можно выразить формулой (см. [28])…”

“…Если D -выпуклое тело, то эта функция является вогнутой (см., например, [29]) на D, а ее супердифференциал в точке x ∈ int D выражает формула (см. [28])…”