The formula for the subdifferential of the distance function to a convex set in an nonsymmetrical space

Abstract

The distance function, defined by the gauge (the Minkowsky gauge function) of a convex body compact, from a point to a convex closed set is considered in a finite-dimensional space. It is known that this function is convex in the whole space. The formula of its the subdifferential is obtained. It includes the subdifferential of gauge function and the cone of feasible directions of set to which the distance is measured, taken in one of the projection points on this set. This circumstans makes it different from the subdifferentional formula received earlier by B. N. Pshenichny in which another characteristics of the objects, defined the distance function, are used. Examples of applications of the obtained formula are given. In particular, a specific form of the subdifferential formula is given for the case when the set, the gauge of which specifies the distance function, and the set to which the distance is measured are lower Lebesgue sets of convex functions.