came known as the KKM principle [12]. The KKM principle inspired countless mathematicians, yielding a formidable body of work in nonlinear and convex analysis; a production known today as the KKM theory. The reader is referred to Dugundji-Granas [7], Park [20] and Yuan [22] for surveys of results, methods, and applications of the KKM theory. 

The particular version of the KKM principle discussed here, which we call the convex KKM principle , is more than sufficient to prove in a direct and economical way, such fundamental results as the Stampacchia Theorem on vari- ational inequalities, the Mazur-Schauder theorem on the minimization of lower semicontinuous quasiconvex and coercive functionals, and the Markov-Kakutani fixed point theorem for commuting families of affine transformations (see e.g., Br´ezis [6]). It is well-known, since Kakutani [16], that the Hahn-Banach theo- rem can be derived from the Markov-Kakutani fixed point theorem. Thus, the equivalence between the Hahn-Banach theorem, Klee’s intersection theorem, the convex KKM principle, and the Markov-Kakutani fixed point theorem is thus established. 

## 2 Preliminaries 

The fundamental tool for our proof of the convex KKM theorem is the separation of a point and a closed convex set in a finite dimensional space. For the sake of completeness, we include the basic separation properties in finite dimensions with the simplest of proofs (see e.g., Magill and Quinzii [19]) 

Lemma 1 Let C be a nonempty closed convex subset of R n and let x / ∈ C. Denote by y = P$_{C}$ ( x ) the projection of x onto C . Then the hyperplane H x C 

