Approximate isoperimetry for convex polytopes
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.