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

Tight Bounds for the Maximum Distance Over a Polytope to a Given Point