Homogeneous polynomial approximation on symmetric convex bodies: A proof of Kroó's conjecture

Abstract

We resolve Kroó's 2004 conjecture on homogeneous polynomial approximation in every dimension, without smoothness or curvature assumptions. For every origin-symmetric convex body $K\subset\mathbb R^d$ with nonempty interior, every continuous real function on $\partial K$ is uniformly approximated by sums of homogeneous polynomials of degrees $n$ and $n+1$ as $n\to\infty$. Moreover, approximation of the constant function by homogeneous polynomials of degree $2n$ is uniform over all such bodies in each fixed dimension. Via homogenization, we also resolve Kroó's 2019 weighted approximation conjecture for every convex admissible weight on Euclidean space. The key construction produces homogeneous polynomials with a common Gaussian boundary profile and localization errors controlled by arbitrary polynomial weights. Its analytic core is a weighted Gaussian approximation theorem: for $d\ge2$, sufficiently broad Gaussians on $\mathbb R^{d-1}$ can be approximated by $e^{-f}F$, where $f$ is nonnegative, convex and Lipschitz with $f(0)=0$, and $F$ is a Schwartz function with Fourier support in an arbitrarily small ball about the origin. Small Fourier support allows Taylor truncation to a lower degree, while the remainder enforces decay through homogenization. We then use positive spherical quadrature and degree raising to yield the full approximation theorem.