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

We establish central limit theorems for the volumes of intersections of Bpn (the unit ball of ℓpn) 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 Bpn, evaluated on a random vector distributed uniformly on the unit sphere.