论文标题
二进制标量产品
Binary scalar products
论文作者
论文摘要
令$ a，b \ subseteq \ mathbb {r}^d $均span $ \ mathbb {r}^d $，以便$ \ langle a，b \ rangle \ in \ in \ {0,1 \} $ holds in b $ in $ b \ in $ b \ in $ a \ in b $中的所有$ a \。我们证明$ | a | \ CDOT | B | \ le（d+1）2^d $。这使我们能够解决Bohn，Faenza，Fiorini，Fisikopoulos，Macchia和Pashkovich（2015）（2015）的猜想。这样的多面有的属性，对于每个定义的超平面$ h $，都有一个平行的超平面$ h'$，因此$ h \ cup h'$都包含所有顶点。作者推测，对于每$ d $二维的2级polytope $ p $，$ p $的顶点的产品和$ p $的面积的数量最多为$ d 2^{d+1} $，我们表明是正确的。
Let $A,B \subseteq \mathbb{R}^d $ both span $\mathbb{R}^d$ such that $\langle a, b \rangle \in \{0,1\}$ holds for all $a \in A$, $b \in B$. We show that $ |A| \cdot |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 $H'$ such that $H \cup H'$ 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.
