On ${\cal Z}_p$-norms of random vectors

TL;DR

This paper investigates bounds on the ${ m Z}_p$-norms of random vectors, proposing a conjecture and establishing results under symmetry conditions, linking geometric and probabilistic estimates.

Contribution

It introduces a conjecture on ${ m Z}_p$-norm bounds for random vectors and proves it under symmetry assumptions, connecting it with covering numbers and Sudakov bounds.

Findings

Conjecture on ${ m Z}_p$-norm bounds formulated.

Proved the conjecture under symmetry assumptions.

Connected the conjecture with covering number estimates.

Abstract

To any $n$-dimensional random vector $X$ we may associate its $L_p$-centroid body ${\cal Z}_p(X)$ and the corresponding norm. We formulate a conjecture concerning the bound on the ${\cal Z}_p(X)$-norm of $X$ and show that it holds under some additional symmetry assumptions. We also relate our conjecture with estimates of covering numbers and Sudakov-type minorization bounds.