Combinatorial slicing problems of polytopes: How (not) to reconstruct a polytope from its slices
Abstract
We study combinatorial aspects of hyperplane sections of polytopes, focusing on how much the combinatorics of the sections determines the combinatorics of the original polytope. We show that, in general, combinatorial information about the sections is not enough to determine even the f-vector of the polytope. In contrast, for sufficiently generic simple polytopes, the function recording the number of vertices of each central section determines the full combinatorial type. We organize different combinatorial slicing properties into hierarchies, separately for affine and central sections, according to the level of combinatorial structure they determine on the polytope. In analogy with classical metric slicing problems, we formulate combinatorial analogues of the Busemann-Petty problem and Bourgain's slicing problem by replacing volume with face numbers, and show that they fail in every dimension and every face dimension.
Related
- Gaussian polytopes with large Banach-Mazur distance to the cross-polytope
- Optimal Slicing Problems
- 0/1 integer programming over polytopes of Chvátal rank one is as hard as lattice problems
- A Combinatorial Benders Decomposition Framework for Two-Dimensional Irregular Bin Packing Problems with Convex Polygons
- On Statistical-Computational Gaps in Graph Combinatorial Testing Problems
BibTeX
@misc{birkemeyer2026combinatorial,
title = {{Combinatorial slicing problems of polytopes: How (not) to reconstruct a polytope from its slices}},
author = {Anna Birkemeyer and Anouk E. Brose and Marie-Charlotte Brandenburg and Niklas Prün},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16195},
eprint = {2609.16195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16195},
}
RIS
TY - UNPB
TI - Combinatorial slicing problems of polytopes: How (not) to reconstruct a polytope from its slices
AU - Anna Birkemeyer
AU - Anouk E. Brose
AU - Marie-Charlotte Brandenburg
AU - Niklas Prün
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16195
UR - https://arxiv.org/abs/2609.16195
AB - We study combinatorial aspects of hyperplane sections of polytopes, focusing on how much the combinatorics of the sections determines the combinatorics of the original polytope. We show that, in general, combinatorial information about the sections is not enough to determine even the f-vector of the polytope. In contrast, for sufficiently generic simple polytopes, the function recording the number of vertices of each central section determines the full combinatorial type. We organize different combinatorial slicing properties into hierarchies, separately for affine and central sections, according to the level of combinatorial structure they determine on the polytope. In analogy with classical metric slicing problems, we formulate combinatorial analogues of the Busemann-Petty problem and Bourgain's slicing problem by replacing volume with face numbers, and show that they fail in every dimension and every face dimension.
ER -