Weak $(1-\epsilon)$-nets for polynomial superlevel sets

TL;DR

This paper establishes the existence of small sets of points in rica such that any polynomial nonnegative on these points has a guaranteed measure of nonnegativity, extending the centerpoint theorem to quadratic and higher-degree polynomials.

Contribution

It introduces new weak fnets for polynomial superlevel sets, generalizing the centerpoint theorem to quadratic and higher-degree polynomials with explicit bounds.

Findings

Existence of small fnets for quadratic polynomials with measure guarantees.

Extension of centerpoint theorem to higher-degree polynomials.

New estimates on Carathe1odory numbers and nonnegative symmetric rank.

Abstract

We prove that for any Borel probability measure $\mu$ on $\mathbb R^n$ there exists a set $X\subset \mathbb R^n$ of $n+1$ points such that any $n$-variate quadratic polynomial $P$ that is nonnegative on $X$ (i.e. $P(x)\geq 0$, for every $x \in X$) satisfies $\mu\{P\geq 0\}\geq \frac{2}{(n+1)(n+2)}$. We also prove that given an absolutely continuous probability measure $\mu$ on $\mathbb R^n$ and $D\leq 2k$, for every $\delta>0$ there exists a set $X\subset \mathbb R^n$ with $|X|\leq \binom{n+2k}{n}-n-1$ such that any $n$-variate polynomial $P$ of degree $D$ that is nonnegative on $X$ satisfies $\mu\{P\geq 0\}\geq \frac{1}{\binom{n+2k}{n}+1} - \delta$. These statements are analogues of the celebrated centerpoint theorem, which corresponds to the case of linear polynomials. Our results follow from new estimates on the Carath\'eodory numbers of real Veronese varieties, or alternatively, from bounds on the nonnegative symmetric rank of real symmetric tensors.