Tight Bounds on the Product of Vertices and Facets for Non-Cube and Non-Cross-Polytope 2-Level Polytopes

Core Concepts

The maximum product of the number of vertices and facets in a 2-level polytope that is not affinely isomorphic to the cube or the cross-polytope is bounded by (d-1) 2^(d+1) + 8(d-1).

Abstract

The content discusses the stability of binary scalar products in 2-level polytopes, which are polytopes where for every facet-defining hyperplane, there is a parallel hyperplane that contains all the vertices of the polytope.
The main results are:

Theorem 2: For a d-dimensional 2-level polytope P that is not affinely isomorphic to the cube or the cross-polytope, the product of the number of vertices (f0(P)) and the number of facets (fd-1(P)) is bounded by (d-1) 2^(d+1) + 8(d-1).

Theorem 3: Let A and B be families of vectors in R^d that both linearly span R^d and have binary scalar products (i.e., the scalar product of any vector in A and any vector in B is either 0 or 1). If |A| and |B| are both at least d+2, then |A| * |B| ≤ d 2^d + 2d.

The proofs of these results build on the previous work by Kupavskii and Weltge, and involve a combination of combinatorial and geometric arguments, including projections onto subspaces and careful case analysis.
The content also discusses examples that demonstrate the tightness of the bounds, as well as a conjecture that generalizes the previous results.

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Key Insights Distilled From

Stability for binary scalar products

by Andrey Kupav... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.17933.pdf

Deeper Inquiries

How can the techniques and insights developed in this work be applied to study the combinatorial structure of other classes of polytopes beyond 2-level polytopes

The techniques and insights developed in this work can be applied to study the combinatorial structure of other classes of polytopes by considering similar properties and constraints. For example, one could investigate polytopes with specific symmetry properties or polytopes that arise in different mathematical contexts. By analyzing the relationships between the number of vertices, facets, and other combinatorial properties, similar stability results and bounds could be derived for these polytopes. Additionally, the approach of considering families of vectors with binary scalar products can be extended to study polytopes with different geometric and combinatorial properties, leading to new insights and results in the field of polyhedral combinatorics.

Are there any connections between the stability of binary scalar products and other areas of mathematics, such as coding theory or discrete geometry

The stability of binary scalar products in the context of polytopes has connections to various areas of mathematics, including coding theory and discrete geometry. In coding theory, binary scalar products are essential in the design and analysis of error-correcting codes, where the properties of binary vectors and their scalar products play a crucial role in encoding and decoding information. By studying the stability of binary scalar products in polytopes, one can potentially derive insights that have applications in coding theory, such as optimizing code performance based on combinatorial structures.
In discrete geometry, the study of binary scalar products is fundamental in understanding the geometric properties of sets of vectors and their relationships. The stability results obtained in this work can provide insights into the combinatorial structure of geometric objects and their interactions, leading to applications in areas such as geometric algorithms, polyhedral theory, and computational geometry. By exploring the connections between binary scalar products and geometric properties, researchers can uncover new connections and applications in diverse areas of mathematics.

Can the conjectured generalization of the main results be proven, and what would be the implications of such a result

The conjectured generalization of the main results could potentially be proven by extending the techniques and proofs developed in this work to accommodate the broader class of polytopes or geometric objects. If the conjecture is proven, it would have significant implications for the understanding of the combinatorial structure of polytopes beyond the 2-level polytopes considered in the current work. The generalization of the main results could lead to a deeper understanding of the relationships between the number of vertices and facets in polytopes with specific properties, providing new insights into the combinatorial complexity of higher-dimensional polytopes.
Furthermore, proving the conjectured generalization could open up new avenues for research in polyhedral combinatorics, discrete geometry, and related fields. The implications of such a result could include the development of more efficient algorithms for polytope enumeration, the discovery of novel geometric properties in higher-dimensional spaces, and the application of combinatorial techniques to solve complex mathematical problems. Overall, proving the conjectured generalization would advance the understanding of polyhedral structures and their combinatorial properties in mathematics.