Quantitative Tverberg theorems over lattices and other discrete sets
  This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantitative versions of Helly's and Carath\'eodory's theorems. 
J. A. De Loera and R. N. La Haye and D. Rolnick and P. Sober\'on
math.MG math.CO