Let M be a subset of Rk. 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(Rn)=n+1 and, by the theorem due to Doignon, Bell and
Scarf, h(Zd)=2d. We give a common extension of these equalities
showing that h(RnΓZd)=(n+1)2d. We show that
the fractional Helly number of the space MβRd (with the
convexity structure induced by Rd) is at most d+1 as long as
h(M) is finite. Finally we give estimates for the Radon number of mixed
integer spaces