The variance conjecture on some polytopes

TL;DR

This paper proves the variance conjecture for random vectors uniformly distributed on certain hyperplane projections of the unit cube and cross-polytope, establishing new cases where the conjecture holds.

Contribution

It demonstrates that vectors on hyperplane projections of $B_1^n$ and $B_^n$ satisfy the variance conjecture, extending known results to these polytopes.

Findings

Variance conjecture verified for hyperplane projections of $B_1^n$ and $B_^n$

Random vectors on these projections satisfy negative square correlation property

Linear images of these vectors also verify the variance conjecture

Abstract

We show that any random vector uniformly distributed on any hyperplane projection of $B_1^n$ or $B_\infty^n$ verifies the variance conjecture $$\text{Var}|X|^2\leq C\sup_{\xi\in S^{n-1}}\E<X,\xi>^2\E|X|^2.$$ Furthermore, a random vector uniformly distributed on a hyperplane projection of $B_\infty^n$ verifies a negative square correlation property and consequently any of its linear images verifies the variance conjecture.