On extremal sections of subspaces of $L_p$

TL;DR

This paper investigates the extremal properties of sections of subspaces of $L_p$ spaces, establishing Schur convexity of certain volume functions, and deriving bounds for projections of these convex bodies.

Contribution

It proves Schur convexity of volume functions for sections of subspaces of $L_p$ and provides new bounds for projections of associated convex bodies.

Findings

The volume function is Schur convex in the squared coordinates.

Minimal volume occurs at equal coordinate weights, maximal at a single coordinate.

Derived bounds for projections of convex bodies in quasi-normed spaces.

Abstract

Let $m,n\in\mathbb{N}$ and $p\in(0,\infty)$. For a finite dimensional quasi-normed space $X=(\mathbb{R}^m, \|\cdot\|_X)$, let $$B_p^n(X) = \Big\{ (x_1,\ldots,x_n)\in\big(\mathbb{R}^{m}\big)^n: \ \sum_{i=1}^n \|x_i\|_X^p \leq 1\Big\}.$$ We show that for every $p\in(0,2)$ and $X$ which admits an isometric embedding into $L_p$, the function $$S^{n-1} \ni \theta = (\theta_1,\ldots,\theta_n) \longmapsto \Big| B_p^n(X) \cap\Big\{(x_1,\ldots,x_n)\in \big(\mathbb{R}^{m}\big)^n: \ \sum_{i=1}^n \theta_i x_i=0 \Big\} \Big|$$ is a Schur convex function of $(\theta_1^2,\ldots,\theta_n^2)$, where $|\cdot|$ denotes Lebesgue measure. In particular, it is minimized when $\theta=\big(\frac{1}{\sqrt{n}},\ldots,\frac{1}{\sqrt{n}}\big)$ and maximized when $\theta=(1,0,\ldots,0)$. This is a consequence of a more general statement about Laplace transforms of norms of suitable Gaussian random vectors which also implies dual estimates for the mean width of projections of the polar body $\big(B_p^n(X)\big)^\circ$ if the unit ball $B_X$ of $X$ is in Lewis' position. Finally, we prove a lower bound for the volume of projections of $B_\infty^n(X)$, where $X=(\mathbb{R}^m,\|\cdot\|_X)$ is an arbitrary quasi-normed space.