One-sided epsilon-approximants

Given a finite point set P⊂R^d, we call a multiset A a one-sided weak ε-approximant for P (with respect to convex sets), if |P∩C|/|P|-|A∩C|/|A|≤ε for every convex set C. We show that, in contrast with the usual (two-sided) weak ε-approximants, for every set P⊂R^d there exists a one-sided weak ε-approximant of size bounded by a function of ε and d.