The distance between two convex sets in Hilbert space

Abstract

In this paper we will discuss how to determine the distance between two convex sets in Hilbert space. This problem came from measuring the shortest distance between two cities, which consider by determining the distance between two buildings in each city. In this problem, the cities are considered as the sets and the buildings are points. Furthermore, based on this problem, it is generalized to determine the distance between two convex sets in Hilbert space that solved by optimization concept by measuring maximal distance between two parallel supporting hyperplanes that separate them. Therefore, it is given some example to understanding, such as the distance between two normed balls, ellipsoids, and linear varieties.