Intersection Theory

Abstract

In this chapter we deal with intersection theory. This theory studies various conditions under which the collection of sets covering a particular set has a nonempty intersection. Such existence results are called intersection theorems, which are often used to derive fixed point theorems. Those theorems are also used to prove the existence of solutions to mathematical programming, game theoretic and equilibrium problems. The most well-known intersection theorem is the lemma of Knaster, Kuratowski and Mazurkiewicz [1929] (KKM lemma) on the unit simplex S ⁿ. This lemma states that n closed subsets covering the (n − 1)-dimensional unit simplex S ⁿ and satisfying some boundary condition have a nonempty intersection. A related theorem was due to Scarf [1967b, 1973]. The reformulation of this theorem was also given by Fan [1968]. Further generalizations of intersection theorems on the unit simplex can be found in Shapley [1973], Freidenfelds [1974], Gale [1984], Freund [1984b], Ichiishi [1988], and Joosten and Tal-man [1993]. The above theorems all build upon bounded sets. Helly [1923] proved the following different type of intersection theorem on an unbounded set. Let C ⁱ, i ∈ I _l be convex sets of ℝⁿ with l ≥ n + 1. If any n + 1 of them have a nonempty intersection, then ∩ ^l _{h =1} C ⁱ ≠ Ø. Helly’s theorem has been generalized in many ways. The interested reader should consult Stoer and Witzgall [1970]. In this chapter we focus on the results along the line of KKM lemma.