A Note on Non-Negative $L_1$-Approximating Polynomials

$L_1$-Approximating polynomials, i.e., polynomials that approximate indicator functions in $L_1$-norm under certain distributions, are widely used in computational learning theory. We study the existence of $\mathit{\text{non-negative}}$ $L_1$-approximating polynomials with respect to Gaussian distributions. This is a stronger requirement than $L_1$-approximation but weaker than sandwiching polynomials (which themselves have many applications). These non-negative approximating polynomials have recently found uses in smoothed learning from positive-only examples. In this short note, we prove that every class of sets with Gaussian surface area (GSA) at most $\mathrm{\Gamma }$ under the standard Gaussian admits degree-$k$ non-negative polynomials that $\text{eps}$-approximate its indicator functions in $L_1$-norm, for $k=\tilde{O}({\mathrm{\Gamma }}^2\mathrm{/}{\epsilon }^2)$. Equivalently, finite GSA implies $L_1$-approximation with the stronger pointwise guarantee that the approximating polynomial has range contained in $[0,\mathrm{\infty })$. Up to a constant-factor, this matches the degree of the best currently known Gaussian $L_1$-approximation degree bound without the non-negativity constraint.