On the Convexified Sauer–Shelah Theorem

LetAbe a subset of {0, 1}n. Givenε>0, we can find a subsetIof {1, ..., n} such that the convex hull in RIof the projection ofAonto {0, 1}Icontains the cube [12−ε, 12+ε]I, and that cardI⩾n−K(nε+n log(2n/card A)), whereK>0 is a universal constant.