Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Given a convex body $K$ of diameter $\Delta$ in $\mathbb{R}^d$ for fixed $d$, the objective is to minimize the number of vertices (alternatively, 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 $O((\Delta/\varepsilon)^{(d-1)/2})$ facets suffice. While this bound is optimal in the case of a Euclidean ball, it is far from optimal for ``skinny'' convex bodies. A natural way to characterize a convex object's skinniness is in terms of its relationship to the Euclidean ball. Given a convex body $K$, define its surface diameter $\Delta_{d-1}$ to be the diameter of a Euclidean ball of the same surface area as $K$. It follows from generalizations of the isoperimetric inequality that $\Delta \geq \Delta_{d-1}$. We show that, under the assumption that the width of the body in any direction is at least $\varepsilon$, it is possible to approximate a convex body using $O((\Delta_{d-1}/\varepsilon)^{(d-1)/2})$ facets. This bound is never worse than the previous bound and may be significantly better for skinny bodies. The bound is tight, in the sense that for any value of $\Delta_{d-1}$, there exist convex bodies that, up to constant factors, require this many facets. The improvement arises from a novel approach to sampling points on the boundary of a convex body. We employ a classical concept from convexity, called Macbeath regions. We demonstrate that Macbeath regions in $K$ and $K$'s polar behave much like polar pairs. We then apply known results on the Mahler volume to bound their number.

翻译：逼近凸体是几何学中的一个基本问题，具有广泛的应用。给定固定维度 $d$ 中直径为 $\Delta$ 的凸体 $K$，目标是在给定 Hausdorff 误差 $\varepsilon$ 下最小化逼近多面体的顶点数（或面数）。Dudley (1974) 提出的已知最优一致界表明，$O((\Delta/\varepsilon)^{(d-1)/2})$ 个面即可满足要求。虽然此界在欧几里得球情况下是最优的，但对于“瘦长”凸体而言远非最优。表征凸体瘦长程度的一个自然方式是其与欧几里得球的关系。给定凸体 $K$，定义其表面直径 $\Delta_{d-1}$ 为与 $K$ 具有相同表面积的欧几里得球的直径。从等周不等式的推广可知 $\Delta \geq \Delta_{d-1}$。我们证明：在假设体在任何方向上的宽度至少为 $\varepsilon$ 的条件下，可用 $O((\Delta_{d-1}/\varepsilon)^{(d-1)/2})$ 个面逼近凸体。此界绝不劣于先前结果，且对瘦长体可能显著更优。该界是紧的，即对于任意 $\Delta_{d-1}$ 值，存在凸体所需面数（至多相差常数因子）达到此下界。这一改进源于一种在凸体边界上采点的新方法。我们采用凸性中的经典概念——Macbeath 区域，证明 $K$ 及其极体内的 Macbeath 区域行为类似于极对，进而应用 Mahler 体积的已知结果来界定其数量。