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

TL;DR

This paper investigates the asymptotic volume of intersections of generalized p-ellipsoids, establishing threshold behaviors and a central limit theorem under spectral flatness conditions, with implications for approximation problems.

Contribution

It introduces a spectral flatness condition to analyze intersection volumes and proves a CLT for p-ellipsoid sampled points at the critical threshold.

Findings

Threshold behavior of intersection volumes characterized

Central limit theorem established for p-ellipsoid samples

Spectral flatness condition determines non-critical behavior

Abstract

Motivated by classical works of Schechtman and Schmuckenschl\"ager 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.