标题： 近似等周不等式对于凸多面体 显示英文标题

标题： Approximate isoperimetry for convex polytopes

摘要： 对于所有具有$\phi\geqslant n+1$的$n,\phi\in \mathbb{N}$，其具有$\phi$个面的$n$维凸多面体的最小可能等周商被证明由$\max\big\{n/\sqrt{1+\log (\phi/n)},\sqrt{n}\big\}$的正的普遍常数倍数上下界所限制。 对于所有$n\in \mathbb{N}$和$2n\leqslant \beta\in 2\mathbb{N}$，证明了每个具有$\beta$个顶点的中心对称凸多面体，其维数为$n$，都存在一个仿射像，其等周商最多是$\min\big\{\sqrt{\log(\beta/n)},n\big\}$的一个通用常数倍，这是精确的。 弱同构反等周猜想对于具有$O(n)$个面的$n$维凸多面体被证明，通过证明任何这样的多面体$K$在体积保持矩阵下的图像$K'$和凸体$L\subseteq K'$，使得$L$的等周商最多是$\sqrt{n}$的一个通用常数倍，并且$\sqrt[n]{\mathrm{vol}_n(L)/\mathrm{vol}_n(K)}$至少是一个正的通用常数。 显示英文摘要

摘要： For all $n,\phi\in \mathbb{N}$ with $\phi\geqslant n+1$, the smallest possible isoperimetric quotient of an $n$-dimensional convex polytope that has $\phi$ facets is shown to be bounded from above and from below by positive universal constant multiples of $\max\big\{n/\sqrt{1+\log (\phi/n)},\sqrt{n}\big\}$. For all $n\in \mathbb{N}$ and $2n\leqslant \beta\in 2\mathbb{N}$, it is shown that every $n$-dimensional origin-symmetric convex polytope that has $\beta$ vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of $\min\big\{\sqrt{\log(\beta/n)},n\big\}$, which is sharp. The weak isomorphic reverse isoperimetry conjecture is proved for $n$-dimensional convex polytopes that have $O(n)$ facets by demonstrating that any such polytope $K$ has an image $K'$ under a volume preserving matrix and a convex body $L\subseteq K'$ such that the isoperimetric quotient of $L$ is at most a universal constant multiple of $\sqrt{n}$, and also $\sqrt[n]{\mathrm{vol}_n(L)/\mathrm{vol}_n(K)}$ is at least a positive universal constant.