Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Consider a convex body $K$ of diameter $\Delta$ in $\textbf{R}^d$ 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 $\varepsilon$. It is known from classical results of Dudley (1974) and Bronshteyn and Ivanov (1976) that $\Theta((\Delta/\varepsilon)^{(d-1)/2})$ vertices (alternatively, facets) are both necessary and sufficient. While this bound is tight in the worst case, that of Euclidean balls, 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 \emph{volume diameter} $\Delta_d$ to be the diameter of a Euclidean ball of the same volume as $K$, and define its \emph{surface diameter} $\Delta_{d-1}$ analogously for surface area. It follows from generalizations of the isoperimetric inequality that $\Delta \geq \Delta_{d-1} \geq \Delta_d$. Arya, da Fonseca, and Mount (SoCG 2012) demonstrated that the diameter-based bound could be made surface-area sensitive, improving the above bound to $O((\Delta_{d-1}/\varepsilon)^{(d-1)/2})$. In this paper, we strengthen this by proving the existence of an approximation with $O((\Delta_d/\varepsilon)^{(d-1)/2})$ facets.

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。考虑固定维度d下直径Δ的凸体K⊂R^d。目标是针对给定Hausdorff误差ε，最小化逼近多面体的顶点数（等价于面数）。由Dudley (1974)和Bronshteyn与Ivanov (1976)的经典结果可知，Θ((Δ/ε)^{(d-1)/2})个顶点（或面）既是必要的也是充分的。虽然该界在欧氏球这一最坏情形下是紧的，但对于细长凸体远非最优。刻画凸体细长性的一种自然方式是借助其与欧氏球的关系。给定凸体K，定义其体积直径Δ_d为与K同体积的欧氏球的直径，类似地定义其表面积直径Δ_{d-1}为与K同表面积的欧氏球的直径。由等周不等式的推广可得Δ ≥ Δ_{d-1} ≥ Δ_d。Arya、da Fonseca和Mount (SoCG 2012)证明了基于直径的界可改进为对表面积敏感，将上述上界提升为O((Δ_{d-1}/ε)^{(d-1)/2})。本文通过证明存在具有O((Δ_d/ε)^{(d-1)/2})个面的逼近，进一步强化了这一结果。