2024/11/02 by Andrés Cristi, Cristi, Andrés, David Pac Salas +1
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory #Metric Geometry (math.MG) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2411.11864
openalex publication_date 2024/11/02 · openalex created_date 2024/11/21 · openalex updated_date 2026/07/28
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 (1)/(e)-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) ≥ (1)/(2n)(1)/(e)Hd(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.