Optimal polynomial meshes exist on any multivariate convex domain

TL;DR

The paper proves that optimal polynomial meshes exist on any convex domain ${\Omega} \subset \mathbb{R}^d$ for all $d \ge 3$, resolving Kroó's conjecture. It develops a Dubiner-type metric $\rho_\Omega$ to quantify polynomial variation, proves a sharp pointwise norm control $|Q(\boldsymbol{x})-Q(\boldsymbol{y})| \le C_* n \rho(\boldsymbol{x},\boldsymbol{y})$ for $Q\in \Pi_n^d$, and reduces the key estimate to a two-dimensional problem via a planar section and affine normalizations. Central to the argument are two pillars: a doubling property for the Dubiner metric and the construction of fast decreasing polynomials, which together bound the number of sample points by a dimension-dependent constant and yield the $N \le C n^d$ mesh. The result provides a practical and geometry-agnostic norming set for discretizing $L_\infty$ norms and underpins applications in discrete approximation and cubature on convex domains.

Abstract

We show that optimal polynomial meshes exist for every convex body in $\mathbb{R}^d$, confirming a conjecture by A. Kroo.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• Proposition 3.2
• Lemma 3.3
• proof