Convex unions and completions from simplicial pseudomanifolds

While intersections of convex sets are convex, their unions have rather complicated behavior. Some natural contexts where they appear include duality arguments involving boundaries of convex sets and valuations, which have an Euler characteristic-like structure. However, there are certain settings where the convexity property itself is important to consider. For example, this includes (preservation of) positivity properties of divisors on toric varieties under blowdowns. In the case of (restrictions of) conormal bundles, this can be interpreted in terms of interactions between local convexity data stored in rational equivalence relations. We consider generalizations to realizations of simplicial pseudomanifolds and replace rational equivalence with effects of PL homeomorphisms. Decomposing the PL homeomorphisms into edge subdivisions and contractions, we characterize the space of suitable contraction points compatible with local convexity properties in terms of convex unions and completions. This gives rise to certain external edge subdivisions that make this contraction space'' of the starting edge empty, which is unexpected given the expected increased convexity’’ from edge subdivisions. We also obtain strong affine/linear restrictions on realizations of facets containing nearby edges preserving local convexity. This implies that contracting certain nearby edges results in a very large or very small contraction space of the starting edge. As for boundary behavior, there are parallels between effects of PL homeomorphisms on induced 4-cycles in the 1-skeleton. Finally, we find effects of PL homeomorphisms and suspensions on analogues of local convexity properties stored by linear systems of parameters. This indicates that simplicial spheres PL homeomorphic to the boundary of a cross polytope store record local convexity changes in the most natural way.

💡 Research Summary

The paper investigates how convexity behaves under unions in the context of realizations of simplicial pseudomanifolds, motivated by applications ranging from duality arguments for convex bodies to positivity of divisors on toric varieties after blow‑downs. The authors replace the rational equivalence relations that appear in toric geometry with the effects of piecewise‑linear (PL) homeomorphisms on simplicial pseudomanifolds. By a classical result of Alexander, any PL homeomorphism can be decomposed into a sequence of edge subdivisions and edge contractions.

The central object introduced is the contraction space of a fixed edge e: the set of points that can serve as the image of e under a PL contraction while preserving the local convexity data encoded by “wall crossings”. Using a hyperplane (non‑separation) interpretation of wall crossings (Proposition 2.2), the authors give a precise description of this space in terms of convex unions and completions (Theorem 3.4, Definition 3.3). An edge subdivision always preserves existing convex wall crossings and creates new ones (Proposition 3.2), but surprisingly, certain external edge subdivisions can make the contraction space of e empty (Theorem 3.20). This counter‑intuitive phenomenon shows that increasing the number of convex wall crossings does not guarantee that more contraction points become admissible.

The paper further derives strong affine/linear restrictions on the realizations of facets that contain edges “nearby” to e. When two facets share the same linear span (i.e., the same hyperplane), the wall crossings between them are linked, and the geometry forces the contraction space of e to collapse either to a single point or to the whole line segment realizing e (Theorem 3.19). These restrictions depend on whether the hyperplanes defining the local convexity half‑spaces separate the endpoints of the edge and on the relative projection lengths (Proposition 3.21).

A parallel line of inquiry examines the effect of PL homeomorphisms on induced 4‑cycles in the 1‑skeleton. The authors show that flat wall crossings correspond precisely to the presence of such 4‑cycles, and PL transformations can create or destroy them (Proposition 3.23). This mirrors phenomena observed in the boundaries of cross‑polytopes, where the 1‑skeleton is covered by induced 4‑cycles.

In the later sections the authors replace rational equivalence with linear systems of parameters (LSPs), which provide linear relations among vertex coordinates (e.g., center‑of‑mass relations or stress spaces). By choosing LSPs adapted to the PL homeomorphisms, they track how local convexity data changes under subdivisions, contractions, and suspensions (Propositions 4.11, 4.12, 4.15). The sign patterns of the coefficients reflect the convex/flat/concave nature of wall crossings, echoing the classical wall relations in toric geometry (Proposition 1.7).

A striking conclusion is that simplicial spheres PL‑homeomorphic to the boundary of a cross‑polytope record changes in local convexity in the most “pure” way: the LSPs on such spheres transform exactly as the underlying convex data, without extra combinatorial interference (Proposition 4.15, Corollary 4.17).

Overall, the work provides a comprehensive framework linking convex geometry, PL topology, and combinatorial algebra. It introduces the novel notion of a contraction space governed by convex unions and completions, establishes affine constraints that dictate when an edge’s contraction space is maximal or minimal, and shows how linear algebraic tools (LSPs) can faithfully encode the evolution of local convexity under PL operations. These insights open avenues for further research on convexity‑preserving transformations of high‑dimensional complexes, optimization on piecewise‑linear manifolds, and the combinatorial foundations of toric positivity.