Distributional stability of the Szarek and Ball inequalities

TL;DR

This paper extends classical inequalities by proving distributional stability results for the Szarek and Ball inequalities, using Fourier analysis to show these inequalities remain nearly optimal under small distributional perturbations.

Contribution

It introduces stability versions of Szarek's Khinchin inequality and Ball's cube slicing inequality for distributions close to Rademacher, expanding their applicability.

Findings

Extended Szarek's inequality to near-Rademacher distributions.

Extended Ball's cube slicing inequality to similar distributions.

Established Fourier-analytic methods for stability analysis.

Abstract

We prove an extension of Szarek's optimal Khinchin inequality (1976) for distributions close to the Rademacher one, when all the weights are uniformly bounded by a $1/\sqrt2$ fraction of their total $\ell_2$-mass. We also show a similar extension of the probabilistic formulation of Ball's cube slicing inequality (1986). These results establish the distributional stability of these optimal Khinchin-type inequalities. The underpinning to such estimates is the Fourier-analytic approach going back to Haagerup (1981).