Approximate isoperimetry for convex polytopes

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