Torricellian points in normed linear spaces

Abstract

Abstract Given a set of n (distinct) points A in a normed space, we consider the set of Torricellian points, that is, the set of points which minimises the sum of distances to the points in A . We introduce the Torricellian functional associated to a set of distinct points A , which calculates the sum of distances of a point x to the points in A . The Torricellian point is defined as the infimum (over all vectors) of this functional. We discuss the existence of Torricellian points in reflexive normed spaces, non-expansive subspaces and evidently, inner product spaces. A case for collinear points is given and is utilised to characterise strict convexity. For a non-collinear case, it is shown that the set of Torricellian points contains a unique point when the space is strictly convex. However, we show that the uniqueness of Torricellian point of a non-collinear set does not characterise strict convexity. We consider a particular example of the Torricellian problem in a space endowed with the Taxicab geometry. MSC:46B20, 49J27.