Theorem 6 (Convex KKM Theorem) Let E be a t.v.s.,  ̸ = X ⊆ Y ⊆ E with Y convex. If Γ : X −→ 2 Y is a set-valued map verifying: 

(i) Γ is a KKM map; 

(ii) all values of Γ are non-empty, closed and convex. Then, the family { Γ( x ) }$_{x}$$_{∈}$$_{X}$ has the finite intersection property. If in addition, there exists a non-empty subset X 0 of X contained in a convex 

compact subset D of Y such that ⋂ x ∈ X 0 Γ( x ) is compact, then ⋂ x ∈ X Γ( x ) ̸ =  . 

Proof. We prove that Proposition 3 is equivalent to Theorem 6. 

(= ⇒ ) Let Γ : X −→ 2 Y be a KKM map with closed convex values. We show by induction on n that conv ( { x 1 , . . . , x$_{n}$ } ) ∩ ⋂ n i $_{=1}$Γ( x$_{i}$ ) ̸ =  , for any finite subset { x 1 , . . . , x$_{n}$ } of X. 

When n = 1 , x 1 = conv ( { x 1 } ) ⊂ Γ( x 1 ) . 

Assume that the conclusion holds true for any set with n = k elements, and let n = k +1 . Put C = conv ( { x 1 , . . . , x$_{n}$ } ) and C$_{i}$ = Γ( x$_{i}$ ) ∩ C. Since Γ is KKM, C ⊆ ⋃ n i $_{=1}$Γ( x$_{i}$ ) which implies C = ⋃ n i $_{=1}$(Γ( x$_{i}$ ) ∩ C ) = ⋃ n i $_{=1}$C$_{i}$ , a convex set. By the induction hypothesis, for each i, we have conv ( { x 1 , . . . , ˆ x$_{i}$, . . . , x$_{n}$ } ) ∩ ⋂ n j =1 ,j ̸ = i Γ( x$_{j}$ ) ̸ =  . Proposition 4 implies that ⋂ n i $_{=1}$(Γ( x$_{i}$ ) ∩ C ) ̸ =  , i.e., ⋂ n i $_{=1}$Γ( x$_{i}$ ) ̸ =  . 

( ⇐ =) Assume C 1 , . . . , C$_{n}$, C = ⋃ n i $_{=1}$C are closed convex sets in a topological vector space satisfying hypotheses (i) and (ii) of Proposition 3 above. 

For each j, let x$_{j}$ ∈ ⋂ n i =1 ,i ̸ = j C$_{i}$ and consider X = { x$_{j}$ } n j $_{=1}$. The set C being convex, conv ( X ) ⊆ C and for all j, i with j ̸ = i, x$_{j}$ ∈ C$_{i}$, which im- plies that A$_{i}$ = conv ( { x$_{j}$ } n j =1 ,j ̸ = $_{i}$) ⊂ C$_{i}$. Define Γ : X −→ 2 C by Γ( x$_{i}$ ) := C$_{i}$ for each i = 1 , ..., n. The values of Γ are clearly closed and convex. Also, conv ( X ) ⊆ C = ⋃ n i $_{=1}$( C$_{i}$ ∩ C ) = ⋃ n i $_{=1}$Γ( x$_{i}$ ) , and for each { x$_{i}$$_{1}$, . . . , x$_{i}$$_{k}$ } ⊂ X, we have conv ( { x$_{i}$$_{1}$, . . . , x$_{i}$$_{k}$ } ) ⊂ A$_{i}$$_{j}$ ⊂ C$_{i}$$_{j}$ = Γ( x$_{i}$$_{j}$ ) for some j ̸ = 1 , . . . , k. Hence 

