Limit theorems for the volumes of small codimensional random sections of ℓ_pⁿ-balls

2022/06/28 by Adamczak, Radosław, Pivovarov, Peter, Simanjuntak, Paul
#52A22 #60D05 #60F05 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.2206.14311

Abstract

We establish Central Limit Theorems for the volumes of intersections of B_pⁿ (the unit ball of ℓ_pⁿ) with uniform random subspaces of codimension d for fixed d and n→ ∞. As a corollary we obtain higher order approximations for expected volumes, refining previous results by Koldobsky and Lifschitz and approximations obtained from the Eldan--Klartag version of CLT for convex bodies. We also obtain a Central Limit Theorem for the Minkowski functional of the intersection body of B_pⁿ, evaluated on a random vector distributed uniformly on the unit sphere.

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