An Elementary Exposition of Pisier's Inequality

TL;DR

This paper offers an accessible, elementary proof of Pisier's inequality, a key result in normed space geometry, and introduces a new restriction on Fourier spectra of functions on the discrete cube.

Contribution

It provides a simplified, constructive proof of Pisier's inequality and explores its applications to Fourier analysis on the discrete cube.

Findings

Elementary proof of Pisier's inequality presented

New Fourier spectrum restriction for functions on the discrete cube

Enhanced accessibility for applications in geometry and analysis

Abstract

Pisier's inequality is central in the study of normed spaces and has important applications in geometry. We provide an elementary proof of this inequality, which avoids some non-constructive steps from previous proofs. Our goal is to make the inequality and its proof more accessible, because we think they will find additional applications. We demonstrate this with a new type of restriction on the Fourier spectrum of bounded functions on the discrete cube.