The Fermat-Weber Problem and Inner-Product Spaces Original Research Article

In the setting of real normed spaces, we study the Fermat-Weber problem which deals with the minimization of the sum of weighted distances from a variable point to the points of a given finite set A. With techniques of best approximation we obtain a description of the set of solutions to this problem. Then we characterize inner product spaces as spaces in which the set of solutions to such problems meets the affine hull of A. The major tool is a characterization of inner product spaces, with finite dimension at least three, lying on some property of the exposed points of the unit ball.