Totally Splittable Polytopes

A split of a polytope is a (necessarily regular) subdivision with exactly two maximal cells. A polytope is totally splittable if each triangulation (without additional vertices) is a common refinement of splits. This paper establishes a complete classification of the totally splittable polytopes.

💡 Research Summary

The paper investigates the class of polytopes whose every triangulation (without adding new vertices) can be obtained as a common refinement of splits, i.e., totally splittable polytopes. A split is a regular subdivision that separates the vertex set into exactly two parts by a single affine hyperplane; the two convex hulls of the parts constitute the only maximal cells. Because splits are the simplest non‑trivial regular subdivisions, a polytope is totally splittable precisely when its secondary fan is generated by the hyperplanes defining all possible splits.

The authors begin by formalising split compatibility: two splits can coexist in a common refinement only if their defining hyperplanes do not intersect in a way that forces a vertex to belong to both sides simultaneously. This condition translates into a partial order on the bipartitions of the vertex set. They then introduce vertex splits (splits that isolate a single vertex) and show that in a totally splittable polytope every vertex must admit a vertex split; otherwise a triangulation lacking that split could not be expressed as a refinement of splits.

The core of the classification relies on Gale duality. By mapping the vertex configuration of a d‑dimensional polytope P to a Gale diagram in ℝ^{n−d−1}, the authors study the combinatorial structure of the set of possible splits. They prove that if every triangulation of P refines splits, the Gale diagram must be one‑dimensional, i.e., all points lie on a line and are partitioned into at most two antipodal clusters. This severe restriction forces the original polytope into one of three families:

1. Simplices – When the Gale diagram consists of a single cluster, P is a d‑simplex. Every split is a vertex split, and the collection of all vertex splits trivially refines any triangulation.

2. Cross‑polytopes – When the Gale diagram consists of two antipodal clusters of equal size, P is a d‑dimensional cross‑polytope (the convex hull of the ± unit vectors). Each coordinate hyperplane yields a split, and all such coordinate splits are mutually compatible, generating the full secondary fan.

3. Prisms over simplices – When the Gale diagram contains two clusters of different cardinalities, the polytope is the Cartesian product of a (d‑1)‑simplex with a line segment, i.e., a prism whose base is a simplex. Splits arise either from the prism direction (separating the two “layers”) or from the vertex splits of the base simplex; together they generate every possible triangulation.

The paper shows that any other polytope necessarily contains a non‑compatible pair of splits or a triangulation that cannot be expressed as a split refinement, thereby violating total splittability. For instance, the 3‑dimensional cube, although a product of two segments, is not a prism over a simplex and fails the condition because its Gale diagram is two‑dimensional.

Finally, the authors discuss implications for the structure of the split complex (the simplicial complex whose vertices are splits and whose faces correspond to compatible families). For totally splittable polytopes, this complex coincides with the entire secondary complex, providing a combinatorial model that is both simple and complete. The classification also yields algorithmic corollaries: recognizing a totally splittable polytope reduces to checking whether the polytope is affinely equivalent to one of the three families, which can be done in polynomial time via facet‑vertex incidence analysis.

In summary, the paper delivers a clean and exhaustive classification: the only totally splittable polytopes are simplices, cross‑polytopes, and prisms over simplices. This result bridges split theory, secondary polytopes, and Gale duality, and it clarifies the exact circumstances under which splits alone suffice to generate all regular triangulations.