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$C_0$的距离问题。假设该多面体外接于已知球体，我们构造了一个保留该球体边界上多面体顶点的球交体，并证明了该球交体可任意精度逼近原多面体。随后，利用关于球交体上最大化到给定点距离的已知结论，构造了一个保留原问题最大值点的新多面体。接着以类似方式得到新的球交体，我们猜想经过有限次迭代后，最终得到的球交体可最大化到给定点的距离。证明所得距离是原距离的非平凡上界。针对维度$n = 100$以内的单位超立方体上到随机点的距离最大化问题进行了测试，并展示了若干详细的二维示例。