Spectral flatness and the volume of intersections of $p$-ellipsoids

Abstract

Motivated by classical works of Schechtman and Schmuckenschläger on intersections of ${\ell }_p$-balls and recent ones in information-based complexity relating random sections of ellipsoids and the quality of random information in approximation problems, we study the threshold behavior of the asymptotic volume of intersections of generalized $p$-ellipsoids. The non-critical behavior is determined under a spectral flatness (Wiener entropy) condition on the semi-axes. In order to understand the critical case at the threshold, we prove a central limit theorem for $q$-norms of points sampled uniformly at random from a $p$-ellipsoid, which is obtained under Noether's condition on the semi-axes.

2nd version with extended examples and their proofs

References in corpus (1)

• A probabilistic approach to the geometry of the \ell_p^n-ball

Cited by in corpus (2)

• Random sections of ${\ell }_p$-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators
• Limit theorems for mixed-norm sequence spaces with applications to volume distribution