Approximating the Euclidean Ball by Polytopes: Surprising Rates in High Dimensions

Abstract

We provide sharp estimates on the number of vertices needed to approximate the Euclidean unit ball $B_d\subset\mathbb R^d$ with relative symmetric-difference volume error at most $\varepsilon\in(0,1)$. We show that the minimum vertex budget $N_\varepsilon(d)$ satisfies $$\log N_\varepsilon(d)\asymp d\log\left(1+\frac1{\varepsilon\sqrt d}\right).$$ In particular, $\exp(O(\sqrt d/\varepsilon))$ vertices suffice for every $\varepsilon\in(0,1)$ and all sufficiently large $d$. We also prove that achieving a fixed error $\varepsilon\in(0,1)$ requires at least $\Omega\bigl((\tfrac{1+\varepsilon}{2\sqrt\varepsilon})^{d-1}\bigr)$ facets, an exponential number in $d$. We extend our vertex approximation result to Gaussian symmetric-difference error, and give a positive answer to the question of De, Nadimpalli, and Servedio about subexponential vertex complexity for approximating Euclidean balls. We also disprove a Macbeath-type extremality principle for unrestricted vertex approximation. Whereas the ball uniformly maximizes the inscribed approximation error among all convex bodies (Macbeath, 1951), we show that approximating the hypercube $[-1,1]^d$ to relative error $\varepsilon$, with $\delta:=1-\varepsilon$, requires $N\ge\frac{\delta}{2}\exp(\Omega(\delta^2d/\log^2(2/\delta)))$ vertices whenever $d\ge16\log(2/\delta)$. Thus, when $d$ is large, the cube is harder to approximate than the ball in this sense. Finally, we determine the sharp dimensional order of the classical Laguerre-Delone triangulation number, showing that $\mathrm{ldel}_{d-1}=\Theta(d^{-1/2})$ as $d\to\infty$. This improves both the upper bound of Ludwig, Sch\"utt, and Werner and the lower bound of B\"or\"oczky by a factor of order $\sqrt d$. By classical asymptotic approximation theorems (Ludwig, 1999), this improvement applies to every convex body in $\mathbb R^d$ with $C^2$ boundary and positive Gaussian curvature.