We study the approximability of general convex sets in $\mathbb{R}^n$ by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution $N(0,I_n)$ and the complexity of an approximation is the number of halfspaces used. While a large body of research has considered the approximation of convex sets by intersections of halfspaces under distance metrics such as the Lebesgue measure and Hausdorff distance, prior to our work there has not been a systematic study of convex approximation under the Gaussian distribution. We establish a range of upper and lower bounds, both for general convex sets and for specific natural convex sets that are of particular interest. Our results demonstrate that the landscape of approximation is intriguingly different under the Gaussian distribution versus previously studied distance measures. For example, we show that $2^{\Theta(\sqrt{n})}$ halfspaces are both necessary and sufficient to approximate the origin-centered $\ell_2$ ball of Gaussian volume 1/2 to any constant accuracy, and that for $1 \leq p < 2$, the origin-centered $\ell_p$ ball of Gaussian volume 1/2 can be approximated to any constant accuracy as an intersection of $2^{\widetilde{O}(n^{3/4})}$ many halfspaces. These bounds are quite different from known approximation results under more commonly studied distance measures. Our results are proved using techniques from many different areas. These include classical results on convex polyhedral approximation, Cram\'er-type bounds on large deviations from probability theory, and -- perhaps surprisingly -- a range of topics from computational complexity, including computational learning theory, unconditional pseudorandomness, and the study of influences and noise sensitivity in the analysis of Boolean functions.

翻译：我们研究了在$\mathbb{R}^n$中一般凸集被半空间交逼近的可能性，其中逼近质量以标准高斯分布$N(0,I_n)$衡量，而逼近的复杂度由使用的半空间数量决定。尽管已有大量研究在勒贝格测度和豪斯多夫距离等距离度量下考虑凸集的半空间交逼近，但在我们这项工作之前，尚未有系统研究高斯分布下的凸逼近问题。我们针对一般凸集及具有特殊意义的特定自然凸集，建立了一系列上界和下界。我们的结果表明，与先前研究的距离度量相比，高斯分布下的逼近景观呈现出令人瞩目的差异。例如，我们证明：对于高斯体积为1/2的以原点为中心的$\ell_2$球，需要且仅需$2^{\Theta(\sqrt{n})}$个半空间即可达到任意常数精度逼近；而对于$1 \leq p < 2$，高斯体积为1/2的以原点为中心的$\ell_p$球，可通过$2^{\widetilde{O}(n^{3/4})}$个半空间的交达到任意常数精度逼近。这些界比常见距离度量下的已知逼近结果有显著不同。我们的结果综合运用了多个不同领域的技术，包括凸多面体逼近的经典结论、概率论中Cramér型大偏差界，以及——或许令人意外——计算复杂性领域的诸多主题，如计算学习理论、无条件伪随机性，以及布尔函数分析中的影响与噪声敏感性研究。