with no loss of generality that { B ( x$_{i}$ ) } n i =1 and { A − $^{1}$( y$_{j}$ ) } m j =1 are minimal covers of M and D respectively. That is, for any k ∈ { 1 , . . . , n } , M ⊈ ⋃ n i =1 ,i = $_{k}$B ( x$_{i}$ ) , and for any l ∈ { 1 , . . ., m } , D ⊈ ⋃ m j =1 ,j = $_{l}$A − $^{1}$( y$_{j}$ ) . Consider the compact convex sets M$_{i}$ = M \ B ( x$_{i}$ ) for for i = 1 , . . . , n, and D$_{j}$ = D \ A − $^{1}$( y$_{j}$ ) for j = 1 , . . . , m. The fact that M ⊆ ⋃ n i $_{=1}$B ( x$_{i}$ ) is equivalent to ⋂ n i $_{=1}$M$_{i}$ =  , and D ⊆ ⋃ m j $_{=1}$A − $^{1}$( y$_{j}$ ) is equivalent to ⋂ m j $_{=1}$D$_{j}$ =  . The minimality of the covers { B ( x$_{i}$ ) } n i =1 and { A − $^{1}$( y$_{j}$ ) } m j =1 amounts to M ∩ ⋂ n i =1 ,i = $_{k}$M$_{i}$ =  for any k ∈ { 1 , . . ., n } , and D ∩ ⋂ m j =1 ,j = $_{l}$D$_{j}$ =  for any l ∈ { 1 , . . ., m } . All conditions of Klee’s theorem (see Remark 4) are thus satisfied for both families of compact convex sets { M, M 1 , . . . , M$_{n}$ } and { D, D 1 , . . . , D$_{m}$ } . Hence, M ⊈ ⋃ n i $_{=1}$M$_{i}$ and D ⊈ ⋃ m j $_{=1}$D$_{j}$ . Let y 0 ∈ M but y 0 / ∈ M$_{i}$ for all i = 1 , . . . , n, and let x 0 ∈ D but x 0 / ∈ D$_{j}$ for all j = 1 , . . . , m. Clearly, y 0 ∈ B ( x$_{i}$ ) ⇐⇒ x$_{i}$ ∈ B − $^{1}$( y 0 ) for all i = 1 , . . . , n, and x 0 ∈ A − $^{1}$( y$_{j}$ ) ⇐⇒ y$_{j}$ ∈ A ( x 0 ) for all for all j = 1 , . . . , m. The sections B − $^{1}$( y 0 ) and A ( x 0 ) being convex sets, it follows that x 0 ∈ D = conv ( { x 1 , . . . , x$_{n}$ } ) ⊂ B − $^{1}$( y 0 ) and y 0 ∈ M = conv ( { y 1 , . . . , y$_{m}$ } ) ⊂ A ( x 0 ) . The proof is finished as ( x 0 , y 0 ) ∈ A ∩ B. This section illustrates how the geometric results in the preceding section, Theo- rems 6, 9, and 12, are key in deriving a number of landmark results in functional analysis. Intersection theorems as well as fixed point and coincidence theorems have analytical formulations as solvability theorems for systems of nonlinear inequalities (see [1, 2, 3, 4, 9]). These analytical formulations are often more practical when it comes to applications. We start with the analytical formula- tion of the convex KKM principle (equivalently, the fixed point theorem for von Neumann relations) and we derive from it, in a simple and straightforward way, two fundamental results. 4 Analytic Formulations and Applications 12