A Geometric Algorithm for Maximizing the Distance over an Intersection of Balls to a Given Point

In this paper the authors propose a polynomial algorithm which allows the computation of the farthest in an intersection of balls to a given point under three additional hypothesis: the farthest is unique, the distance to it is known and its magnitude is known. As a use case the authors analyze the subset sum problem SSP(S,T) for a given $S\in \mathbb{R}^n$ and $T \in \mathbb{R}$. The proposed approach is to write the SSP as a distance maximization over an intersection of balls. It was shown that the SSP has a solution if and only if the maximum value of the distance has a predefined value. This together with the fact that a solution is a corner of the unit hypercube, allows the authors to apply the proposed geometry results to find a solution to the SSP under the hypothesis that is unique.