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

TL;DR

This paper develops a method to approximate the maximum distance from a point to a polytope using intersections of balls, providing tight bounds and iterative improvements, demonstrated on high-dimensional hypercubes.

Contribution

It introduces a novel iterative approach using intersections of balls to bound and approximate the maximum distance over a polytope, which was not previously available.

Findings

The method provides non-trivial upper bounds for the maximum distance.

The approach is effective up to 100 dimensions in hypercube examples.

Detailed 2D examples illustrate the method's application.

Abstract

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.