论文标题
强烈$ {\ mathbb c} $的卷近似 - 随机polyhedra的凸域
Volume Approximation of Strongly ${\mathbb C}$-Convex Domains by Random Polyhedra
论文作者
论文摘要
第二作者最近考虑了$ \ mathbb {c}^d $中类似凸状域的多面体型近似。特别是，已经获得了最佳体积近似中误差的衰减率，这是相位数量的函数。在本文中，我们通过研究强烈$ \ mathbb {c} $ - 凸域的边界上的随机点（泊松或二项式过程）构建的多面体来进一步进行这些研究。我们通过随机多面体确定域的体积近似误差率，并猜测最小限制常数的精确值。类似于实际情况，在随机体积近似的错误率中出现的指数与最佳体积近似的指数一致，并且可以根据自然存在的度量空间的Hausdorff维度来解释。此外，限制常数被认为取决于莫比乌斯 - 弗弗曼的度量，这是Blaschke表面积测量的复杂类似物。最后，我们还证明了$ l^1 $ - convergence，方差界限和正常近似。
Polyhedral-type approximations of convex-like domains in $\mathbb{C}^d$ have been considered recently by the second author. In particular, the decay rate of the error in optimal volume approximation as a function of the number of facets has been obtained. In this article, we take these studies further by investigating polyhedra constructed using random points (Poisson or binomial process) on the boundary of a strongly $\mathbb{C}$-convex domain. We determine the rate of error in volume approximation of the domain by random polyhedra, and conjecture the precise value of the minimal limiting constant. Analogous to the real case, the exponent appearing in the error rate of random volume approximation coincides with that of optimal volume approximation, and can be interpreted in terms of the Hausdorff dimension of a naturally-occurring metric space. Moreover, the limiting constant is conjectured to depend on the Möbius-Fefferman measure, which is a complex analogue of the Blaschke surface area measure. Finally, we also prove $L^1$-convergence, variance bounds, and normal approximation.
