Art Galleries, k-modems, and k-convexity

Abstract

In 1973, Victor Klee posed the following question: How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls?

This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated a plethora of fun papers, surveys and a book, most of them written in the last twentyfive years. Several variations of the original Art Gallery Problem have appeared, and continue to appear in the literature. In this talk, we will present some recent work motivated by the following problem.

Experience dictates that while trying to locate the best location for a wireless within a building, the main factor that attenuates the signal of a wireless modem, is the number of walls that a signal has to cross. Thus we call a wireless modem (from now on a modem) a k-modem if the signal it transmits is capable of crossing k-walls of a building, and still provide a strong enough signal.

A generalization of Klee’s question is thus: How many k-modems are necessary, and how many are sufficient to cover the interior of an art gallery with n-walls? For k = 0, our problem reduces to the original Art Gallery Problem.