Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls

TL;DR

This paper proves Central Limit Theorems for the volumes of small codimensional sections of $\,ell_p^n$-balls, providing refined asymptotic approximations and extending previous results in convex geometry.

Contribution

It establishes new CLTs for volumes of intersections of $\,ell_p^n$-balls with random subspaces and for the Minkowski functional of intersection bodies, improving existing approximation methods.

Findings

CLTs for volumes of small codimensional sections of $\,ell_p^n$-balls.

Higher order volume approximation results.

CLT for Minkowski functional of intersection bodies.

Abstract

We establish Central Limit Theorems for the volumes of intersections of $B_{p}^n$ (the unit ball of $\ell_p^n$) with uniform random subspaces of codimension $d$ for fixed $d$ and $n\to \infty$. 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 $B_p^n$, evaluated on a random vector distributed uniformly on the unit sphere.