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

Abstract

We establish central limit theorems for the volumes of intersections of ${\mathit{B}}_{\mathit{p}}^{\mathit{n}}$ (the unit ball of ${\ell }_{\mathit{p}}^{\mathit{n}}$) with uniform random subspaces of codimension d for fixed d and $\mathit{n}\to \infty$. 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 ${\mathit{B}}_{\mathit{p}}^{\mathit{n}}$, evaluated on a random vector distributed uniformly on the unit sphere.