A refinement of a fundamental intersection theorem of V. L. Klee for families of closed convex subsets of R n (see Klee [17] and Berge [5]) plays a crucial role in our proof. We provide here a simple proof based on Proposition 2. 

Topological vector spaces (t.v.s. for short), as well as topological spaces, are assumed to be Hausdorff (T 2 ). Vector spaces are assumed real (or complex) and the convex hull of a subset A of a vector space is denoted by conv ( A ) . 

Proposition 3 ([11]) Let C 1 , . . . , C$_{n}$, be non empty closed convex sets in a t.v.s. E such that: 

(i) C = ⋃ n i $_{=1}$C$_{i}$ is convex, and 

(ii) each k of them, 1 ≤ k < n, have a common point. Then ⋂ n i $_{=1}$C$_{i}$ ̸ =  . 

Proof. The proof goes along the lines of Klee’s proof [17]. One may assume with no loss of generality that the sets C$_{i}$, i = 1 , . . . , n, are compact convex subsets of a finite dimensional space. Indeed, one could consider the convex finite polytope ˆ C := Conv ( { y$_{j}$ : j = 1 , . . . , n } ) , where, for each j = 1 , · · · , n, the points y$_{j}$ ∈ ⋂ n i =1 ,i ̸ = j C$_{i}$ are provided by (ii), and define ˆ C$_{i}$ := C$_{i}$ ∩ ˆ C. Clearly, all the sets ˆ C 1 , . . . , ˆ C$_{n}$, ˆ C = ⋃ n i =1 ˆ C$_{i}$ are compact convex sets in a finite dimensional subspace of E and ⋂ n i $_{=1}$C$_{i}$ ̸ =  ⇐⇒ ⋂ n i =1 ˆ C$_{i}$ ̸ =  . 

If n = 1 , the thesis clearly holds. Assume, for a contradiction that for n ≥ 2 , ⋂ n i $_{=1}$C$_{i}$ =  and let us show that (i) must fail if (ii) holds true. The proof is by induction on n. 

If n = 2 , (ii) asserts that both C 1 and C 2 are nonempty and, while they are disjoint, their union C = C 1 ∪ C 2 cannot be convex and thus (i) fails. 

Suppose that for n = k − 1 , it holds ( ⋂ k − 1 i =1 C$_{i}$ =  and ⋂ k − 1 i =1 ,i ̸ = j C$_{i}$ ̸ =  ) = ⇒ ⋃ k − 1 i =1 C$_{i}$ is not convex. k 

Let n = k, and let { C$_{i}$ } i =1 be a collection of compact convex sets such that C$_{k}$ ∩ ⋂ k − 1 i =1 C$_{i}$ =  and ∀ j = 1 , ..., k, ⋂ k i =1 ,i ̸ = j C$_{i}$ ̸ =  . By Proposition 

