Transversal numbers over subsets of linear spaces

Abstract

Let M be a subset of R-k. It is an important question in the theory of linear inequalities to estimate the minimal number h = h(M) such that every system of linear inequalities which is infeasible over M has a subsystem of at most h inequalities which is already infeasible over M. This number h(M) is said to be the Helly number of M. In view of Helly's theorem, h(R-n) = n+1 and, by the theorem due to Doignon, Bell and Scarf, h(Z(d)) = 2(d). We give a common extension of these equalities showing that h(R-n x Z(d)) = (n + 1)2(d). We show that the fractional Helly number of the space M subset of R-d (with the convexity structure induced by R-d) is at most d+1 provided h(M) is finite. Finally we give estimates for the Radon number of mixed integer spaces.