Limit theorems for the volumes of small codimensional random sections of ${\mathrm{\ell }}_p^n$-balls

We establish Central Limit Theorems for the volumes of intersections of $B_p^n$ (the unit ball of ${\mathrm{\ell }}_p^n$) with uniform random subspaces of codimension $d$ for fixed $d$ and $n\to \mathrm{\infty }$. 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.