In 1960, B. Grünbaum proved that, for any convex body \(C\subset \mathbb {R}^d\) and every halfspace H containing the centroid of C, the volume of \(H\cap C\) is at least a \(\frac{1}{e}\) -fraction of the volume of C. In 2014, Oertel conjectured that a similar result holds for mixed-integer convex sets. Concretely, he proposed that for any convex body \(C\subset \mathbb {R}^{n+d}\) , there should exist a point \(\textbf{x} \in S=C\cap (\mathbb {Z}^{n}\times \mathbb {R}^d)\) such that every halfspace H containing \(\textbf{x}\) satisfies \( \mathop {\mathcal {H}}\nolimits _d(H\cap S) \ge \frac{1}{2^n}\frac{1}{e}\mathop {\mathcal {H}}\nolimits _d(S), \) where \(\mathcal {H}_d\) denotes the d-dimensional Hausdorff measure. While the conjecture remains open, Basu and Oertel proved in 2017 that the above inequality holds for sets that are sufficiently large in terms of a measure known as the lattice width. In this work, we improve upon this result, 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, thereby significantly enlarging the family of mixed-integer convex sets for which Oertel’s conjecture holds.
