On Extremal Sections of Subspaces of $L_p$

Abstract

Let $m,n\in {\mathbb {N}}$ and $p\in (0,\infty )$. For a finite dimensional quasi-normed space $X=({\mathbb {R}}^m, \Vert \cdot \Vert _X)$, let

$$\begin{aligned} B_p^n(X) = \left\{ (x_1,\ldots ,x_n)\in \big ({\mathbb {R}}^{m}\big )^n: \sum _{i=1}^n \Vert x_i\Vert _X^p \leqslant 1\right\} . \end{aligned}$$

We show that for every $p\in (0,2)$ and X which admits an isometric embedding into $L_p$, the function

$$\begin{aligned} S^{n-1} \ni \uptheta = (\uptheta _1,\ldots ,\uptheta _n) \longmapsto \left| B_p^n(X) \cap \left\{ (x_1,\ldots ,x_n)\in \big ({\mathbb {R}}^{m}\big )^n: \sum _{i=1}^n \uptheta _i x_i=0 \right\} \right| \end{aligned}$$

is a Schur convex function of $(\uptheta _1^2,\ldots ,\uptheta _n^2)$, where $|\cdot |$ denotes Lebesgue measure. In particular, it is minimized when $\uptheta =\big (\frac{1}{\sqrt{n}},\ldots ,\frac{1}{\sqrt{n}}\big )$ and maximized when $\uptheta =(1,0,\ldots ,0)$. 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 $(B_p^n(X))^\circ$ 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 $X=({\mathbb {R}}^m,\Vert \cdot \Vert _X)$ is an arbitrary quasi-normed space.