Almost norming vertices for homogeneous polynomials on the cube

Abstract

Let $m\ge 1$ be fixed. We consider the space ${\mathcal{P}}_m({\mathbb{R}}^n)$ of real $m$-homogeneous polynomials on ${\mathbb{R}}^n$, endowed with the standard Gaussian measure ${\gamma }_{m,n}$ associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of ${\ell }_{\infty }^n$, namely the cube $[-1,1{]}^n$, can be recovered from its values at the vertices. For $P\in {\mathcal{P}}_m({\mathbb{R}}^n)$, set \[ M(P)=\max_{x\in[-1,1]^n}|P(x)|, \qquad V(P)=\max_{\varepsilon\in\{-1,1\}^n}|P(\varepsilon)|. \] If $P_n$ is chosen according to ${\gamma }_{m,n}$, we prove that the relative loss \[ 1-\frac{V(P_n)}{M(P_n)} \] is of order at most $n^{-1/2}$ in probability. Consequently, for every $0<\beta <1/2$, \[ γ_{m,n}\left\{ P\in\mathcal{P}_m(\mathbb{R}^n): V(P)\geq (1-n^{-β})M(P) \right\} \longrightarrow1 \] as $n\to \infty$. Thus, with respect to the Bombieri Gaussian measure, the vertices of the cube are asymptotically norming for homogeneous polynomials of fixed degree.

arXiv comment15 pages