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.

翻译：本文提出一种多项式算法，可在三个附加假设下计算球交集中到给定点的最远点：最远点唯一、其距离已知且其模长已知。作为应用实例，作者分析了给定$S\in \mathbb{R}^n$和$T \in \mathbb{R}$的子集和问题SSP(S,T)。所提出的方法是将SSP表述为球交集中距离最大化问题。研究表明，SSP有解当且仅当距离最大值达到预定值。结合解为单位超立方体的顶点这一事实，作者运用所提出的几何结果，在解唯一的假设下找到了SSP的解。