Let $\mathcal{A},\mathcal{B}\subseteq {\mathbb{R}}^d$ both span ${\mathbb{R}}^d$ such that $〈a,b〉\in \{0,1\}$ holds for all $a\in \mathcal{A}$, $b\in \mathcal{B}$. We show that $|\mathcal{A}|\cdot |\mathcal{B}|\le (d+1)2^d$. This allows us to settle a conjecture by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) concerning 2-level polytopes. Such polytopes have the property that for every facet-defining hyperplane H there is a parallel hyperplane $H^{\prime }$ such that $H\cup H^{\prime }$ contain all vertices. The authors conjectured that for every d-dimensional 2-level polytope P the product of the number of vertices of P and the number of facets of P is at most $d{2}^{d+1}$, which we show to be true.

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二进制标量产品

让$\mathcal{一种},\mathcal{乙}\subseteq {\mathbb{R}}^d$两个跨度${\mathbb{R}}^d$这样$〈\mathrm{一种},b〉\in \{0,1\}$适用于所有人$\mathrm{一种}\in \mathcal{一种}$,$b\in \mathcal{乙}$. 我们表明$|\mathcal{一种}|\cdot |\mathcal{乙}|\le (d+1)2^d$. 这使我们能够解决 Bohn、Faenza、Fiorini、Fisikopoulos、Macchia 和 Pashkovich（2015）关于 2 级多面体的猜想。这样的多面体具有这样的性质：对于每个定义面的超平面H，都有一个平行的超平面$H^{\text{'}}$这样$H\cup H^{\text{'}}$包含所有顶点。作者推测，对于每个d维 2 级多面体P ， P的顶点数与P的面数的乘积最多为$d{2}^{d+1}$，我们证明这是真的。