Reducing the large set threshold for Oertel's conjecture on the mixed-integer volume

• 成果类型: Article; Early Access
• 署名作者: Cristi, Andres; Salas, David
• 署名单位: Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Universidad de O'Higgins
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-026-02391-9
• 发表日期: 2026-07-07
• 关键词: convex-bodies; THEOREM

摘要

In 1960, B. Gr & uuml;nbaum proved that, for any convex body C subset of Rd and every halfspace H containing the centroid of C, the volume of H boolean AND C is at least a 1e -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 of Rn+d , there should exist a point x is an element of S=C boolean AND(Zn & times;Rd) such that every halfspace H containing x satisfies where Hd 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.