In this paper the problem of maximizing the distance to a given fixed point over an intersection of balls is considered. It is known that this problem is NP complete in the general case, since any subset sum problem can be solved upon solving a maximization of the distance over an intersection of balls to a point inside the convex hull. The general context is: in [1] it is shown that exists a polynomial algorithm which always solves the maximization problem if the given point is outside the convex hull of the centers of the balls. Naturally one asks if there is a polynomial algorithm which solves the problem for a point inside the convex hull. A conjecture stated in a previous paper, [1] is proved, under slightly stronger conditions. The proven conjecture allows a polynomial algorithm for points on the facets of the convex hull and shows that such points share the maximizer with all the points in a small enough ball centered at it, thus including points in the interior of the convex hull of the ball centers.

翻译：本文考虑在球交上最大化到给定固定点的距离问题。已知该问题在一般情况下是NP完全的，因为任何子集和问题都可以通过求解凸包内一点到球交的最大距离来求解。总体背景是：文献[1]证明，若给定点位于球心凸包外部，则存在一个多项式算法总能解决该最大化问题。自然会问：是否存在针对凸包内部点的多项式算法？本文在略强条件下证明了先前论文[1]中提出的猜想。该猜想允许对凸包表面上的点使用多项式算法，并表明这类点与以其为中心的足够小球内的所有点共享最大化点，从而包含球心凸包内部的点。