2022/11/11 by Daniel Galicer, Galicer, Daniel, Alexander E. Litvak +5
Biochemistry, Genetics and Molecular Biology · Mathematics · #52A20 (secondary) #52A21 #52A23 #52A38 #52A40 (primary) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Point processes and geometric inequalities #Prion Diseases and Protein Misfolding
paper · pdf · doi:10.48550/arxiv.2211.06094
openalex publication_date 2022/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body K we show that there is another convex body L such that the volume ratio between any two projections of fixed rank of the bodies K and L is large. Namely, we prove that for every 1≤ k≤ n and for each convex body K⊂ ℝn there is a centrally symmetric body L ⊂ ℝn such that for any two projections P, Q: ℝn → ℝn of rank k one has vr(PK, QL) ≥ c min\( k)/( √(n)) √((1)/(log log log((nlog(n))/(k)))), (√(k))/(√(log((nlog(n))/(k))))\, where c>0 is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime k≥ n2/3.