Doignon proved a discrete version of Helly's theorem
claiming that a finite family of convex sets in R^n intersects in an
integral point if every subfamily of size at most 2^n does so.
Motivated by applications in integer programming, Aliev et al.
recently obtained a quantitative version of this result, which
guarantees that a finite family of convex sets intersects in k
integral points whenever every subfamily of size at most c_n(k) does
so. The best current upper bound on the minimal such constant c_n(k)
grows linearly with the parameter k. Based on a connection to the
number of boundary integral points in strictly convex sets, we show
that the asymptotic behavior of c_n(k) is sublinear in dimension two
and we determine the exact value of c_n(k) for k at most four.
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