Optimal Area-Sensitive Bounds for Polytope Approximation

Approximating convex bodies is a fundamental problem in geometry. Given a convex body $K$ in $\mathbb{R}^d$ for a fixed dimension $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff error $\varepsilon$. The best known uniform bound, due to Dudley (1974), shows that $O((\text{diam}(K)/\varepsilon)^{(d-1)/2})$ facets suffice. Although this bound is optimal for fat objects, such as Euclidean balls, it is far from optimal for ``skinny'' convex bodies. Skinniness can be characterized relative to the Euclidean ball. Given a convex body $K$, define its area radius, $\text{arad}(K)$, to be the radius of the Euclidean ball having the same surface area as $K$. It follows from generalizations of the isoperimetric inequality that $\text{diam}(K) \geq 2 \cdot \text{arad}(K)$. We show that, given a convex body whose minimum width is at least $\varepsilon$, it is possible to approximate the body by a polytope having $O((\text{arad}(K)/\varepsilon)^{(d-1)/2})$ facets. Our approach works by first reducing the problem of approximating convex bodies to that of approximating convex functions. We employ a classical concept from convexity, called Macbeath regions. We demonstrate that there is a polar relationship between the Macbeath regions of a function and the Macbeath regions of its Legendre dual. This is combined with known bounds on the Mahler volume to bound the total size of the approximation.