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(B) there exists ¯ x ∈ Y such that f (¯ x, y ) ≤ λ, for all y ∈ Y. Consequently, when λ = sup$_{x}$$_{∈}$$_{X}$ f ( x, x ) , (A) is impossible and 

Proof. Let A ( x ) := { y ∈ Y : f ( x, y ) > λ } , x ∈ X. All hypotheses of Theorem 9 are satisfied except, possibly, the non-emptiness of the sections A ( x ) . Thus, either A ( x ) ̸ =  , ∀ x ∈ X, hence A is a von Neumann relation, and therefore has a fixed point ((A) holds), or A (¯ x ) =  for some ¯ x ∈ X, i.e., A is not a von Neumann relation and (B) is satisfied. 

This is a particular instance of the celebrated I nfsup Inequality of Ky Fan with a weaker compactness condition. 

Landmark theorems of nonlinear functional analysis follow immediately from Theorem 14; therefore, indirectly, from the separation of closed convex sets in finite dimension (Proposition 2). We refer to H. Br´ ezis [6] for an account and applications of the next two fundamental results. 

Corollary 15 (Mazur-Schauder Theorem) Let X be a non-empty closed convex subset of a reflexive Banach space E and let ϕ : X −→ R be a lower semicon- tinuous, quasiconvex and coercive (i.e. lim$_{|}$$_{|}$$_{x}$$_{|}$$_{|}$$_{→∞}$ ϕ ( x ) = ∞ ) functional. Then ϕ achieves its minimum on X. 

Proof. Let λ = 0 , Y = X, and f ( x, y ) = ϕ ( x ) − ϕ ( y ) in Theorem 14. Let K be the intersection of X with a closed ball with radius M > 0 centered at the origin of E and such that if x ∈ X with ‖ x ‖ > M then ϕ ( x ) > ϕ ( y ) for some y ∈ K. Such a non-empty set K exists due to the coercivity of ϕ. Since E is reflexive, K is weakly compact. One readily verifies that the hypotheses of Theorem 14 with X, Y, f, K, D = K, and λ = 0 all hold: f is l.s.c. and quasiconvex in x, and quasiconcave in y. Clearly, possibility (A) of Theorem 14 cannot hold. Hence 

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