Distributional stability of the Szarek and Ball inequalities

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 $1/\sqrt{2}$ 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).

Final version. To appear in Math. Ann

References in corpus (5)

• Transport-majorization to analytic and geometric inequalities
• Slicing ${\ell }_p$-balls reloaded: stability, planar sections in ${\ell }_1$
• Extremal sections and projections of certain convex bodies: a survey
• Khinchin-type inequalities via Hadamard's factorisation
• Sharp bounds on $p$-norms for sums of independent uniform random variables, $0<p<1$

Cited by in corpus (2)

• Stability of polydisc slicing
• Resilience of cube slicing in ${\ell }_p$