In this paper we show that Minty's lemma can be used to prove the Hahn-Banach theorem as well as other theorems in this class such as Radon's and Heiiy's theorem for oriented matroids having an intersection property which guarantees that every pair of flats intersects in some point extension 6 Up of the oriented matroid C.
In
[12] Las Vergnas introduced the notion of convexity for oriented matroids in order to study analogues of the Hahn-Banach theorem. Cordovil [5,6] proved versions of the Hahn-Banach theorem for oriented matrcds of rank three. As Mandel [13] pointed out this theorem is no longer true for oriented matroids of rank greater than three. In this paper we show that Minty's lemma can be used to prove a slightly stronger form of the Hahn-Banach theorem as well as other theorems in this class su& as Radon's and Helly's theorem for oriented matroids of any rank provided the oriented matroids have a so-called intersection property. This intersection property (IP) came across when investigating polars of oriented matroids. It defines a class of oriented matroids which is contained in the more general class of Euclidezg matroids and which includes those oriented matroids having aa oriented adjoint (i.e. allowing the construction of polar@. Loosely speaking the intersection property guarantees that every pair of flats intersects in some p&t extension 6 Up of 0. In case of unoriented matroids of rank 4 this is equivalent to what geometers call the bundle condition (cfi Kern IlOT, .I -Ler '@ be an oriented matroid. We consider the circuits of 0 as vectors X of 2 fE ::a. I+ -, O}E and as usual write 8(X', X-, x") for the set of all e E E for which X,;O (Xc= +, Xe= -, Xc = 0) and also use the notation XA 2 0 (~0) if ) respectively X 3 O(SOj if A = E. A vector X E 2*E is called a cell of 6 (resp. cocell) if X is an element of the circuit span (resp. cocircuit span) of 0. Note that a vector of X E 2*E is a cocell of 6 if and only if it is orthogonal to all circuits C of 0. For the sake of simplicity we sha