Approximate isoperimetry for convex polytopes

2025/09/17 by Ball, Keith, Böröczky, Károly J., Naor, Assaf
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2509.13898

Abstract

For all n,ϕ∈ ℕ with ϕ\geqslant n+1, the smallest possible isoperimetric quotient of an n-dimensional convex polytope that has ϕ facets is shown to be bounded from above and from below by positive universal constant multiples of max\n/√(1+log (ϕ/n)),√(n)\. For all n∈ ℕ and 2n\leqslant β∈ 2ℕ, it is shown that every n-dimensional origin-symmetric convex polytope that has β vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of min\√(log(β/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' under a volume preserving matrix and a convex body L⊆ K' such that the isoperimetric quotient of L is at most a universal constant multiple of √(n), and also √[n]vol_n(L)/vol_n(K) is at least a positive universal constant.

Citations

• The separation modulus of unitarily invariant matrix norms
• An integer parallelotope with small surface area
• Volume ratios and a reverse isoperimetric inequality
• Shadows of convex bodies

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