Cone and constrained colorful Carathéodory Theorems

Abstract: Holmsen proved in 2016 a generalization of the classical colorful Caratheodory theorem in which a matroid imposes additional constraints on the desired colorful transversal. His approach also works in the more general setting of oriented matroids, rather than relying directly on convex hulls. In this paper, we extend these ideas in several directions. First, we study which colorful Caratheodory-type results remain valid when convex cones replace convex hulls, as well as analogous modifications in the oriented matroid setting. Second, we consider variants in which the additional constraint on the transversal is not encoded by a matroid. This leads to new extensions of the classical Tverberg theorem. Our approach is topological, following the methods of Holmsen, and Kalai and Meshulam, on which it builds. The key idea is to analyze homology groups of simplicial complexes that encode colorful Caratheodory-type phenomena, such as the support complex of an oriented matroid. In particular, one shows that these complexes are (near-)d-Leray. We extend this analysis by carrying out more detailed homology computations for these complexes, with the aim of enabling further and more refined applications of the method.

• The paper extends classical Carathéodory theorems by replacing convex hulls with cone operators and incorporating generalized constraints.
• It employs refined homological methods on support complexes, leading to new selection theorems even when traditional Leray conditions fail.
• The work provides explicit counterexamples and structural insights, advancing applications in algorithmic convexity and topological data analysis.

Cone and Constrained Colorful Carathéodory Theorems: Generalizations and Topological Methods

Introduction

The paper "Cone and constrained colorful Carathéodory Theorems" (2607.06172) systematically advances the theory of combinatorial convexity, focusing on colorful Carathéodory-type results under both conic and constrained conditions. Building on the interleaving strands of classical convexity, matroid theory, and oriented matroids, the authors develop several new theorems that extend the reach of traditional results. These generalizations are motivated by both the affine and the topological perspectives and are achieved through refined homological methods applied to support complexes associated with various combinatorial abstractions of convexity.

Background, Previous Generalizations, and Context

The original colorful Carathéodory theorem of Bárány and its classical relatives (e.g., Tverberg's theorem) form cornerstones of combinatorial convex geometry. The Bárány theorem ensures, under specific convex intersection conditions of color classes, the existence of a "colorful" selection such that the origin lies in their convex hull. Over the past decades, a series of generalizations have been developed, incorporating matroid constraints [Kalai, Meshulam; Holmsen], alternative operators like convex cones [Bárány], and abstractions to oriented matroid settings.

Key points of prior generalizations include:

• Partition Matroid Generalization: Affine convex hull replaced by matroid independence and the existence of supporting convex intersections [Kalai, Meshulam, Holmsen].
• Oriented Matroid Generalization: Affine convexity statements are phrased combinatorially in terms of positive circuits in oriented matroids, thus broadening geometric intuitions to purely combinatorial settings.
• Conic Variant: The convex hull is replaced by the cone generated by colored sets, leading to new existence results for generating prescribed elements by conical combinations.

Holmsen's 2016 extension in particular developed a unified topological approach, analyzing (near-)$d$-Leray properties of the associated support complexes, allowing for the powerful use of the Homological Nerve Theorem.

Main Results and New Generalizations

This paper presents several core advancements:

• Replacement of Convex Hull by Cone Operators: The authors characterize the settings in which colorful Carathéodory-type results hold when convex hulls are replaced by convex cones, both in affine and oriented matroid frameworks. This yields genuinely new types of selection theorems.
• Removal of Matroidal Constraints: The authors establish that certain generalizations hold even when the constraint on the transversal is encoded by more general types of simplicial complexes rather than matroids (specifically, certain joins of color classes replaced by connected subcomplexes).
• Refined Homology Computations: By performing detailed homological analyses of the aforementioned support complexes, the paper obtains existence results even in frameworks where original Leray-based methods do not apply directly and enables new applications to previously open cases.

One highlight is the constrained colorful Carathéodory theorem, which posits the existence of a face in a constrained join of color classes with the desired containment property, even when the join is not a matroid independence complex but a more general connected subcomplex.

Another is a tight oriented matroid cone version:

Let $\mathscr{O}$ be an oriented matroid of rank $\leq d$ over $V \sqcup \{v_0\}$, and $\mathscr{M}$ a matroid on $V$. If for all $U \subseteq V$ with $\rho_{\mathscr{M}}(V - U) < d$ we have $v_0 \in \operatorname{conv}(U)$ (interpreted in $\mathscr{O}$), then there is an independent $\mathscr{O}$0 of $\mathscr{O}$1 with $\mathscr{O}$2.

It is proven that these conic analogues do not directly generalize their convex hull counterparts, nor do they always imply each other—evidenced by explicit counterexamples and homological obstructions.

The theorems are unified in a schematic "meta-theorem"—showing systematic logical dependencies between them and clarifying which classical forms imply others via combinatorial substitutions.

Methodology and Topological Approach

The authors extend and repurpose the topological framework of Helly-type and Carathéodory-type theorems. The central methodological ideas are:

• Support Complex Analysis: Given a convexity (or matroid) predicate $\mathscr{O}$3 on subsets of a ground set $\mathscr{O}$4, the "avoiding complex" $\mathscr{O}$5 is studied as an abstract simplicial complex. Its (near-)Leray property underpins the method.
• Homological Nerve Theorem: The authors exploit that if a suitable candidate complex $\mathscr{O}$6 (often a join or other prescribed structure) cannot be embedded in a near-$\mathscr{O}$7-Leray complex, a desired selection exists.
• Detailed (Co)homological Computations: The support, vector-avoiding, and element-avoiding complexes' homology and links are computed explicitly in terms of pseudosphere arrangements, affording much sharper analyses than mere Leray-vanishing.

Notably, the approach is robust to non-matroidal choices of constraints, so long as homological connectivity of the candidate complex is sufficiently high. This delivers: new Carathéodory theorems, analysis of variants that fail, and pathways for generalizations.

Strong Claims, Counterexamples, and Theoretical Implications

The paper provides explicit counterexamples to demonstrate:

• Limits of Conic Generalizations: Cone versions of certain results fail in settings ($\mathscr{O}$8 colors) where their convex hull counterparts succeed.
• Structural Independence: Certain constrained combinatorial Carathéodory theorems cannot be derived from matroidal forms or from classical oriented matroid versions; they genuinely require novel structural insights.
• Homological Barriers: The failure of colorful selection in some cases can be traced to the non-vanishing of key homology groups or lack of required connectivity in the avoidance complexes.

It also proves strong new positive results, such as:

• Extensions to Ordered and Constrained Tverberg Partitions: The constrained colorful theorem yields strengthened Tverberg-type intersection properties for partitions with additional combinatorial structure—potentially of use in future combinatorial convexity and topological data analysis contexts.

Practical and Theoretical Implications

On the practical side, these theorems may inform:

• Algorithmic Convexity: They provide certificates guaranteeing selections in more general combinatorial geometries (e.g., with matroid or even weaker constraints), informing robust convex hull or conic hull selection algorithms in high-dimensional data.
• Topological Data Analysis: The tie to Leray properties and nerve theorems makes these results relevant for guarantees in persistent homology computations of intersection structures in datasets modeled combinatorially.

Theoretically, they represent significant refinements in:

• Combinatorial Geometry: Expanding the world of Helly-type and Carathéodory-type theorems beyond matroidal and affine settings.
• Topological Methods in Discrete Mathematics: Illustrating how a careful homological analysis, even beyond simple vanishing results, can yield new theorems with precise constraints.

Directions for Future Research

The explicit homology computations may be leveraged in even more general settings (e.g., higher cohomological obstructions), in spectral sequence methods, or in applications to mixed convex-geometric and algebraic-combinatorial problems. Developing further non-matroidal constraint theorems and classifying precisely which combinatorial complexes allow for existence results remains a promising direction.

Conclusion

This work substantially extends the landscape of colorful Carathéodory-type theorems to both cone-based and non-matroidally constrained settings, grounded in detailed homological analysis and careful handling of combinatorial abstractions of convexity. The results highlight subtle differences between convex hull and conic hull variants, limit cases for transversals under various constraints, and provide tools and methodology ripe for further generalizations and applications in geometry, topology, and discrete mathematics.