On finding a guard that sees most and a shop that sells most 1

We consider two problems where our goal is to find a point x such that the area of the region V (x) “controlled” by x is as large as possible. In the first problem, we are given a simple polygon P , and V (x) is the visibility polygon of x, that is, the region of points y inside P such that the segment xy does not intersect the boundary of P . In the second problem, we are given a set of points T , and V (x) is the Voronoi cell of x in the Voronoi diagram of the set T ∪ {x}, that is, the set of points that are closer to x than to any point in T . In both problems, it is straightforward (but tedious) to write a closed formula describing the area of the region controlled by a point x. This area function (inside a region where V (x) has the same combinatorial structure) is the sum of the areas of triangles that depend on the location of x. The function domain consists of a polynomial number of regions, and the function has a different closed form in each region: it is the sum of Θ(n) low-degree rational functions in two variables, which do not have common denominator. It seems difficult to solve the problem of finding the maximum of this function analytically and efficiently, and we resort to approximation. In this paper we address the question of efficiently finding a point x that approximately maximizes the area of V (x). More precisely, let μ(x) be the area of V (x), and let μopt = maxx μ(x) be the area for the optimal solution. Given δ > 0, we show how to find xapp such that μ(xapp) ≥ (1− δ)μopt.