Approximate isoperimetry for convex polytopes

Abstract For all with , the smallest possible isoperimetric quotient of an ‐dimensional convex polytope that has facets is shown to be bounded from above and from below by positive universal constant multiples of . For all and , it is shown that every ‐dimensional origin‐symmetric convex polytope that has vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of , which is sharp. The weak isomorphic reverse isoperimetry conjecture is proved for ‐dimensional convex polytopes that have facets by demonstrating that any such polytope has an image under a volume‐preserving matrix and a convex body such that the isoperimetric quotient of is at most a universal constant multiple of , and also is at least a positive universal constant.