Convex order and heat flow for projection profiles of $\ell_p^n$ balls

TL;DR

This paper investigates the distributional properties of projection profiles of $ ext{ell}_p^n$ balls, establishing convex order comparisons, heat-flow regularizations, and extremal coordinate configurations.

Contribution

It introduces a distributional comparison framework for projection profiles, leveraging heat-flow identities to identify coordinate maximizers and diagonal minimizers.

Findings

Convex order comparison based on majorization of squared coordinates.

Heat-flow regularization leads to strict Schur convexity of the profile.

Identification of coordinate maximizers and diagonal minimizers for all positive times.

Abstract

Let $B_p^n$ be the unit ball of $\ell_p^n$, with $1\le p<2$. We study central densities of one-dimensional marginals of the uniform measure on $B_p^n$ and of its Gaussian heat-flow regularizations. The profile is standardized by multiplying the central density by the standard deviation of the marginal. The key comparison is distributional: if the squared coordinates of one direction majorize those of another, then the corresponding squared projection is larger in convex order. A heat-flow identity turns this distributional comparison into strict Schur convexity of the smoothed central profile at every positive time. Together with the classical central-section theorem at $t=0$, this gives coordinate maximizers and diagonal minimizers for every $t\ge0$. We also evaluate the endpoint constants along the standard coordinate-to-diagonal chain and give a fourth-cumulant criterion for monotonicity of the coordinate profile.