0 = ⇒ x ∈ A − $^{1}$( y$_{i}$ ) ⇐⇒ y$_{i}$ ∈ A ( x ) . The section A ( x ) being convex, the convex combination s ( x ) ∈ A ( x ) ∩ P. Theorem 12 (Coincidence ( N , N − $^{1}$) ) Let X and Y be two non-empty convex subsets in topological vector spaces, and let A ∈ N ( X, Y ) , B ∈ N − $^{1}$( X, Y ) . If one of the following compactness conditions holds: (i) Y is compact; or (ii) X is compact; or (iii) there exist a compact subset K of X and a compact convex subset C of Y with A ( x ) ∩ C =  , ∀ x ∈ X \ K ; then A ∩ B =  . Proof. Let it be made clear, first, that ( i ) = ⇒ ( iii ) and ( ii ) = ⇒ ( iii ) . Indeed, if Y is compact, take C = Y and K =  in ( iii ) . If X is compact, take K = X and ( iii ) is vacuously satisfied. Moreover, due to Lemma 11, ( iii ) can be reduced to ( i ). Indeed, assume that ( iii ) holds. Lemma 11 implies the existence of a convex finite polytope P ⊂ Y and a continuous mapping s : K −→ Y with s ( x ) ∈ A ( x ) ∩ P for all x ∈ K. Now, if x ∈ X \ K, A ( x ) ∩ C =  where C ⊂ Y is convex and compact . Consider the convex hull ˆ Y = conv ( P ∪ C ), a compact subset of Y (the convex envelope of two compact convex sets in a topological vector space is also compact convex). It is clear that the map ˆ A : X −→ 2 ˆ Y given by ˆ A ( x ) := A ( x ) ∩ ˆ Y , x ∈ X, defines a von Neumann relation, i.e., ˆ A ∈ N ( X, ˆ Y ) . Also, the mapping ˆ B : X −→ 2 ˆ Y given by ˆ B ( x ) := B ( x ) ∩ ˆ Y , x ∈ X, verifies ˆ B ∈ N − $^{1}$( X, ˆ Y ) . A coincidence for the pair ( ˆ A, ˆ B ) is also a coincidence for ( A, B ) . It suffices, thus, to show that A ∩ B =  under hypothesis ( i ) , i.e., when Y is compact. Since ∀ y ∈ Y, B − $^{1}$( y ) =  , it follows that { B ( x ) }$_{x}$$_{∈}$$_{X}$ forms an open cover of Y. Similarly, { A − $^{1}$( y ) }$_{y}$$_{∈}$$_{Y}$ is an open cover of X. Let { B ( x$_{i}$ ) } n i =1 be a finite subcover of Y, let D := conv ( { x 1 , . . ., x$_{n}$ } ) , a convex compact subset of X, and let { A − $^{1}$( y$_{j}$ ) } m j =1 be an open subcover of D. Consider the convex compact subset M := conv ( { y 1 , . . . , y$_{m}$ } ) of Y. M can be covered by a subfamily of { B ( x$_{i}$ ) } n i $_{=1}$, which, for simplicity we also denote { B ( x$_{i}$ ) } n i $_{=1}$. We can assume 11