[Paper Review] The convex dimension of hypergraphs and the hypersimplicial Van Kampen-Flores Theorem

Leonardo Martínez-Sandoval, Arnau Padrol|arXiv (Cornell University)|Sep 3, 2019

Topological and Geometric Data Analysis42 references3 citations

TL;DR

This paper completely determines the convex dimension of complete k-uniform hypergraphs, proving that it is 2k for n ≥ 2k + 2, n−2 for n ∈ {2k−1, 2k, 2k+1}, and 2n−2k for n ≤ 2k−2. The authors reframe the problem using affine projections that preserve the vertices of the hypersimplex ∆n,k, leading to a hypersimplicial generalization of the van Kampen-Flores theorem and providing a full characterization of projections preserving the i-skeleton of the hypersimplex.

ABSTRACT

The convex dimension of a $k$-uniform hypergraph is the smallest dimension $d$ for which there is an injective mapping of its vertices into $\mathbb{R}^d$ such that the set of $k$-barycenters of all hyperedges is in convex position. We completely determine the convex dimension of complete $k$-uniform hypergraphs, which settles an open question by Halman, Onn and Rothblum, who solved the problem for complete graphs. We also provide lower and upper bounds for the extremal problem of estimating the maximal number of hyperedges of $k$-uniform hypergraphs on $n$ vertices with convex dimension $d$. To prove these results, we restate them in terms of affine projections that preserve the vertices of the hypersimplex. More generally, we provide a full characterization of the projections that preserve its $i$-dimensional skeleton. In particular, we obtain a hypersimplicial generalization of the linear van Kampen-Flores theorem: for each $n$, $k$ and $i$ we determine onto which dimensions can the $(n,k)$-hypersimplex be linearly projected while preserving its $i$-skeleton. Our results have direct interpretations in terms of $k$-sets and $(i,j)$-partitions, and are closely related to the problem of finding large convexly independent subsets in Minkowski sums of $k$ point sets.

Motivation & Objective

• To determine the convex dimension of complete k-uniform hypergraphs K(k)n for all k and n.
• To establish tight bounds on the maximum number of hyperedges in k-uniform hypergraphs with convex dimension d.
• To generalize the linear van Kampen-Flores theorem to hypersimplices by characterizing projections preserving the i-skeleton of ∆n,k.
• To connect convex embeddings to Minkowski sums of point sets and k-sets in discrete geometry.
• To resolve open problems in convex combinatorial optimization and extremal combinatorics related to convex position in high-dimensional configurations.

Proposed method

• Reformulate convex embeddings of hypergraphs as affine projections that strictly preserve the vertices of the hypersimplex ∆n,k.
• Use Ziegler’s projection lemma and Sanyal’s framework to analyze when projections preserve the i-skeleton of ∆n,k.
• Characterize the minimal dimension d(n,k,i) for which ∆n,k admits a projection preserving its i-skeleton.
• Establish a bijection between convex embeddings of complete k-uniform hypergraphs and projections of ∆n,k that preserve its 0-skeleton (vertices).
• Leverage polyhedral geometry and Minkowski sum structures to relate convex embeddings to k-sets and (i,j)-partitions.
• Apply results from reduced plabic graphs and Grassmannian stratifications to support the theoretical framework.

Experimental results

Research questions

• RQ1 What is the exact convex dimension of the complete k-uniform hypergraph K(k)n for all n and k?
• RQ2 For which dimensions d can the (n,k)-hypersimplex be linearly projected while preserving its i-skeleton?
• RQ3 How does the extremal problem of maximizing hyperedges in k-uniform hypergraphs with convex dimension d scale with n and k?
• RQ4 What is the relationship between convex embeddings and the structure of Minkowski sums of k point sets?
• RQ5 Can the van Kampen-Flores theorem be generalized to hypersimplices in a topological or combinatorial sense?

Key findings

• The convex dimension of K(k)n is 2k when n ≥ 2k + 2, n−2 when n ∈ {2k−1, 2k, 2k+1}, and 2n−2k when n ≤ 2k−2, for 2 ≤ k ≤ n−2.
• For k = 1, cd(K(1)n) = 1 if n = 2 and 2 for n ≥ 3; for k = n−1, cd(K(n−1)n) = 2 for n ≥ 3.
• The minimal dimension d(n,k,i) for which ∆n,k admits a projection preserving its i-skeleton is 2k+2i if n ≥ 2k+2i+2, 2n−2k+2i if n ≤ 2k−2i−2, n−1 if 2k−2i−1 ≤ n ≤ 2k+2i+1 and k ∈ An,i, and n−2 otherwise.
• The result provides a hypersimplicial generalization of the linear van Kampen-Flores theorem, determining the minimal dimension for skeleton-preserving projections of hypersimplices.
• The characterization implies that all k-barycenters of an n-point set can be vertices of the k-set polytope if and only if the convex dimension of K(k)n is realized in the corresponding dimension.
• The framework extends to other hypergraphs via their associated hyperedge polytopes, particularly matroid polytopes, and connects to extremal problems in Minkowski sums and convex independence.