Approximate isoperimetry for convex polytopes

Abstract

For all $n,\ddot{I}•\in \mathbb{N}$ with $\ddot{I}•⩾n+1$, the smallest possible isoperimetric quotient of an $n$-dimensional convex polytope that has $\ddot{I}•$ facets is shown to be bounded from above and from below by positive universal constant multiples of $\max \{n/\sqrt{1+\log (\ddot{I}•/n)},\sqrt{n}\}$. For all $n\in \mathbb{N}$ and $2n⩽{\hat{I}}^2\in 2\mathbb{N}$, it is shown that every $n$-dimensional origin-symmetric convex polytope that has ${\hat{I}}^2$ vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of $\min \{\sqrt{\log ({\hat{I}}^2/n)},n\}$, 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 $K^{\prime }$ under a volume preserving matrix and a convex body $L\subseteq K^{\prime }$ 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.