AI 中文总结
该研究将Bárány等人的高斯方差下界扩展到旋转不变对数凹分布生成的随机多面体，通过重心坐标替代处理结合局部方差估计，建立了内蕴体积和面对数的方差下界。
AI 中文摘要
我们针对由ℝᵈ上具有全支撑的旋转不变对数凹概率测度的独立样本生成的随机多面体的内蕴体积和面对数，建立了方差下界。我们的结果将Bárány与Vu、Bárány与Thäle的对应高斯下界扩展到了更广泛的这类分布。我们的证明基于Bárány与Vu引入的几何构造，对此我们开发了基于重心坐标的替代处理方法，并将其与内蕴体积和面对数的局部方差估计相结合。
英文摘要
We establish variance lower bounds for the intrinsic volumes and the face numbers of random polytopes, generated by independent samples from rotationally invariant log-concave probability measures on $\mathbb{R}^d$ with full support. Our results extend the corresponding Gaussian lower bounds of Bárány and Vu and of Bárány and Thäle to this broader class of distributions. Our proof builds on the geometric construction introduced by Bárány and Vu, for which we develop an alternative treatment based on barycentric coordinates. We combine it with local variance estimates for both intrinsic volumes and the number of faces.
CommentsAdded references and revised the discussion of Corollary 1.3
