In this paper we study the problem of maximizing the distance to a given point $C_0$ over a polytope $\mathcal{P}$. Assuming that the polytope is circumscribed by a known ball we construct an intersection of balls which preserves the vertices of the polytope on the boundary of this ball, and show that the intersection of balls approximates the polytope arbitrarily well. Then, we use some known results regarding the maximization of distances to a given point over an intersection of balls to create a new polytope which preserves the maximizers to the original problem. Next, a new intersection of balls is obtained in a similar fashion, and as such, after a finite number of iterations, we conjecture, we end up with an intersection of balls over which we can maximize the distance to the given point. The obtained distance is shown to be a non trivial upper bound to the original distance. Tests are made with maximizing the distance to a random point over the unit hypercube up to dimension $n = 100$. Several detailed 2-d examples are also shown.

翻译：本文研究在多面体$\mathcal{P}$上最大化到给定点$C_0$距离的问题。假设该多面体被一个已知球体所外接，我们构造了一个球体交集，该交集保留了该球体边界上多面体的顶点，并证明该球体交集可以任意逼近多面体。随后，利用已知的关于在球体交集上最大化到给定点距离的结论，构造出一个新的多面体，该多面体保留了原问题的最优解。接着，以类似方式得到一个新的球体交集，经过有限次迭代后，我们推测得到一个可计算到给定点最大距离的球体交集。所得距离被证明是原距离的一个非平凡上界。本文对维度$n=100$以内的单位超立方体上随机点到给定点的距离最大化问题进行了测试，并展示了若干二维详细示例。