Convex bodies with few faces

1989/10/26 by Keith Ball, Alain Pajor, Ball, Keith +1
Mathematics · #10E05 #52A20 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.math/9201203

openalex publication_date 1989/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is proved that if u₁,…, u_n are vectors in \Bbb R^k, k≤ n, 1 ≤ p < ∞ and r = (1\over k ∑ ⁿ₁ |u_i|^p)^{1\over p} then the volume of the symmetric convex body whose boundary functionals are ± u₁,…, ± u_n, is bounded from below as |\ x∈ \Bbb R^k\colon |⟨ x,u_i⟩ | ≤ 1 \hboxfor every i\|^{1\over k} ≥ 1\over √ρr. An application to number theory is stated.

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