Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $\epsilon> 0$, a covering is a collection of convex bodies whose union covers $K$ such that a constant factor expansion of each body lies within an $\epsilon$ expansion of $K$. Coverings have been employed in many applications, such as approximations for diameter, width, and $\epsilon$-kernels of point sets, approximate nearest neighbor searching, polytope approximations, and approximations to the Closest Vector Problem (CVP). It is known how to construct coverings of size $n^{O(n)} / \epsilon^{(n-1)/2}$ for general convex bodies in $\textbf{R}^n$. In special cases, such as when the convex body is the $\ell_p$ unit ball, this bound has been improved to $2^{O(n)} / \epsilon^{(n-1)/2}$. This raises the question of whether such a bound generally holds. In this paper we answer the question in the affirmative. We demonstrate the power and versatility of our coverings by applying them to the problem of approximating a convex body by a polytope, under the Banach-Mazur metric. Given a well-centered convex body $K$ and an approximation parameter $\epsilon> 0$, we show that there exists a polytope $P$ consisting of $2^{O(n)} / \epsilon^{(n-1)/2}$ vertices (facets) such that $K \subset P \subset K(1+\epsilon)$. This bound is optimal in the worst case up to factors of $2^{O(n)}$. As an additional consequence, we obtain the fastest $(1+\epsilon)$-approximate CVP algorithm that works in any norm, with a running time of $2^{O(n)} / \epsilon ^{(n-1)/2}$ up to polynomial factors in the input size, and we obtain the fastest $(1+\epsilon)$-approximation algorithm for integer programming. We also present a framework for constructing coverings of optimal size for any convex body (up to factors of $2^{O(n)}$).

翻译：凸体覆盖已成为设计涉及凸体近似问题高效解的核心组成部分。直观上，给定凸体$K$和$\epsilon>0$，覆盖是指一组凸体的集合，其并集覆盖$K$，使得每个凸体经过常数因子膨胀后仍位于$K$的$\epsilon$膨胀内。覆盖已被广泛应用于许多领域，例如点集直径、宽度和$\epsilon$-核的近似、近似最近邻搜索、多面体近似以及最近向量问题（CVP）的近似。已知对于$\textbf{R}^n$中的一般凸体，可构造规模为$n^{O(n)} / \epsilon^{(n-1)/2}$的覆盖。在特殊情形下（如凸体为$\ell_p$单位球），该界限已改进为$2^{O(n)} / \epsilon^{(n-1)/2}$。这引发了一个问题：此类界限是否普遍成立？本文对此问题给出了肯定回答。我们通过将覆盖应用于Banach-Mazur度量下的凸体多面体近似问题，展示了其强大功能与普适性。给定一个中心对称凸体$K$和近似参数$\epsilon>0$，我们证明存在由$2^{O(n)} / \epsilon^{(n-1)/2}$个顶点（面）构成的多面体$P$，使得$K \subset P \subset K(1+\epsilon)$。在最坏情形下，该界限在$2^{O(n)}$因子意义下是最优的。作为附加结果，我们得到了在任意范数下运行最快的$(1+\epsilon)$-近似CVP算法，其运行时间为$2^{O(n)} / \epsilon^{(n-1)/2}$（忽略输入规模的多项式因子），并得到了整数规划最快的$(1+\epsilon)$-近似算法。我们还提出了一个框架，可为任意凸体构建最优规模的覆盖（在$2^{O(n)}$因子意义下）。