具有固定数量k-面的凸多面体对欧氏球的对偶体积逼近
Dual volume approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces
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中文总结 AI 辅助
该研究延续凸多面体对欧氏球内蕴体积逼近的工作，发展对偶径向理论，得到内接、外接凸多面体体积亏格、平均宽度过剩的非渐近下界，恢复了已有顶点情况的下界阶并推广至所有q∈ℝ。
中文摘要 AI 辅助
我们研究具有指定数量k维面的凸多面体对欧氏球的对偶体积逼近，这延续了作者此前关于具有固定数量k-面的凸多面体对球的内蕴体积逼近的工作，并发展了相应的对偶径向理论。我们的第一个主要结果是，对于满足0≤k≤⌊d/2⌋的情况，内接于欧氏球B_d的凸多面体P_M（其至多有M个k-面）的体积亏格给出非渐近下界，该估计形式为：vol_d(B_d∖P_M)≥(1−e⁻¹)/2 κ_d min{1, (d/2)(ω_d/(4κ_{d−1}))^(2/(d−1)) M^(-2/(d−1))}，在顶点情况k=0且M很大时，该下界恢复了Gordon、Reisner和Schütt（《近似理论杂志》，1997年）所得下界的阶。我们还证明了，对于满足⌈d/2⌉−1≤k≤d−1的情况，至多有M个k-面的外接凸多面体的平均宽度过剩的极对偶结论。更一般地，利用Besau、Hoehner和Kur（《国际数学研究通讯》，2021年）引入的对偶体积偏差的解析延拓，我们在内接和外接两种模型中，对所有实数q（包括q=0）都得到了非渐近下界。
英文摘要
We study dual volume approximation of the Euclidean ball by polytopes with a prescribed number of $k$-dimensional faces. This continues the authors' previous work on intrinsic volume approximation of the ball by polytopes with a fixed number of $k$-faces, and develops the corresponding dual radial theory. Our first main result gives a nonasymptotic lower bound for the volume deficit of an inscribed polytope $P_M\subset B_d$ with at most $M$ $k$-faces, for $0\leq k\leq \lfloor d/2\rfloor$. The estimate has the form \[\operatorname{vol}_d(B_d\setminus P_M) \geq \frac{1-e^{-1}}{2}κ_d\min\left\{ 1,\frac{d}{2}\left(\frac{ω_d}{4κ_{d-1}}\right)^{\frac{2}{d-1}} M^{-\frac{2}{d-1}}\right\},\] and, in the vertices case $k=0$, in the large-$M$ regime it recovers the order of the lower bound of Gordon, Reisner and Schütt (J. Approx. Theory, 1997). We also prove the polar counterpart for the mean width excess of circumscribed polytopes with at most $M$ $k$-faces, for $\lceil d/2\rceil-1\leq k\leq d-1$. More generally, using the analytic extension of the dual volume deviations introduced by Besau, Hoehner and Kur (Int. Math. Res. Not., 2021), we obtain nonasymptotic lower bounds for all $q\in\mathbb R$, including $q=0$, in both the inscribed and circumscribed models.
