The expansion of a polytope is an important parameter for the analysis of the random walks on its graph. A conjecture of Mihai and Vazirani states that all $0/1$-polytopes have expansion at least 1. We show that the generalization to half-integral polytopes does not hold by constructing $d$-dimensional half-integral polytopes whose expansion decreases exponentially fast with $d$. We also prove that the expansion of half-integral zonotopes is uniformly bounded away from $0$. As an intermediate result, we show that half-integral zonotopes are always graphical.

翻译：多面体的扩张性质是分析其图上随机游走的重要参数。Mihai与Vazirani猜想所有$0/1$多面体的扩张至少为1。我们通过构建扩张随维度$d$呈指数衰减的$d$维半整数多面体，证明该结论对半整数多面体的推广并不成立。同时我们证明半整数区域多面体的扩张一致地远离0。作为中间结果，我们证得半整数区域多面体始终具有图结构性。