Abstract

In 1960, Grünbaum proved that for any convex body C⊂Rd and every halfspace H containing the centroid of C, one has that the volume of H∩C is at least a 1e-fraction of the volume of C. Recently, in 2014, Oertel conjectured that a similar result holds for mixed-integer convex sets. Concretely, he proposed that for any convex body C⊂Rn+d, there should exist a point x∈S=C∩(Zn×Rd) such that for every halfspace H containing x, one has that Hd(H∩S)≥12n1eHd(S),where Hd denotes the d-dimensional Hausdorff measure. While the conjecture remains open, Basu and Oertel proved in 2017 that the above inequality holds true for sufficiently large sets, in terms of a measure known as the lattice width of a set. In this work, by following a geometric approach, we improve this result by substantially reducing the threshold at which a set can be considered large. We reduce this threshold from an exponential to a polynomial dependency on the dimension, therefore significantly enlarging the family of mixed-integer convex sets over which Oertel’s conjecture holds true. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.