MINIMAX THEOREM
Minimax Theorem - Proven by von Neumann in 1928, this is a cornerstone of duality (and of game theory). It states that there exists x in S_m and y in S_n such that x'Ay is a saddlepoint of the bilinear form:
Min(Max u'Av: v in S_n): u in S_m) = Max(Min u'Av: u in S_m): v in S_n) = x'Ay.
This extends to the following.
Let F:X*Y-->R be such that X and Y are non-empty, convex , compact sets, F(.,y) is convex on X for each y in Y, and F(x,.) is concave on Y for each x in X. Then, there exists a saddlepoint, (x*,y*) in X*Y: Min(Max F(x,y): y in Y): x in X) = Max(Min F(x,y): x in X): y in Y) = F(x*,y*).
