核心概念
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).
要約
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.