The Cramer-Rao inequality for star bodies
The Cramer-Rao inequality for star bodies
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DOI：
10.1215/s0012-9074-02-11212-5
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发表时间：
2002-03
影响因子：
2.5
通讯作者：
E. Lutwak;Deane Yang;Gaoyong Zhang
中科院分区：
文献类型：
--
作者：
E. Lutwak;Deane Yang;Gaoyong Zhang
与欧几里德 n 空间 R 中的每个物体 K 相关联的是一个椭球 Γ2K，称为 K 的勒让德椭球。它可以定义为以物体质心为中心的唯一椭球，使得该椭球关于穿过质心的任何轴的转动惯量与物体的转动惯量相同。在早期的论文中，作者表明，对应于每个凸体 K ⊂ R 的是一个新的椭球 Γ−2K，它在某种意义上与勒让德椭球对偶。勒让德椭球是对偶 Brunn-Minkowski 理论的一个对象，而新椭球 Γ−2K 是 Brunn-Minkowski 理论的对应对象。本文有两个目标。首先是证明 Γ−2 的定义域可以扩展到星形集。二是证明两个椭球之间存在如下关系：若K是星形集合，则 Г−2K ⊂ Г2K，当且仅当 K 是一个以原点为中心的椭球时，相等。这种包含是信息论基本不等式之一——克莱默-拉奥不等式的几何模拟。与欧几里得 n 空间 R 中的每个物体 K 相关联的是一个椭球 Γ2K，称为 K 的勒让德椭球。勒让德椭球是经典力学的基本概念。它可以定义为以物体质心为中心的唯一椭球体，且该椭球体绕任何通过质心的轴的转动惯量与物体的转动惯量相同。在[26]中，作者表明，对应于每个凸体 K ⊂ R 的是一个新的椭球 Γ−2K。本文的结果暗示了这个新椭球和勒让德椭球之间存在显着的二元性。本文有两个目标。首先是证明 Γ−2 的定义域可以扩展到星形集。二是证明两个椭球之间存在如下关系：若K是星形集合，则(1) Г−2K ⊂ Г2K，当且仅当K是一个以原点为中心的椭球时，相等。这种包含是信息论基本不等式之一——克莱默-拉奥不等式的几何模拟。 Brunn-Minkowski 理论（通常称为混合体积理论）是解析凸几何的核心。该理论的许多基本要素都是由闵可夫斯基在一个世纪前提出的。 Schneider 的书 [28] 是该主题的经典参考书。多年来，布伦-闵可夫斯基理论的工具已被证明在解决数据涉及凸体投影的反演问题方面非常有效。四分之一个世纪前，[22]（以及相关论文）中介绍了对偶 Brunn-Minkowski 理论的要素。对偶理论的基本思想是取代1991年数学学科分类的预测。 52A40、94A17。 1 2 ERWIN LUTWAK、DEANE YANG 和 GAOYON ZHANG 的 Brunn-Minkowski 交集理论。在[23]中表明，这些理论之间实际上存在一个“字典”。不仅经典理论中的“初等混合体积”等概念成为对偶理论中的“初等对偶混合体积”，甚至布伦-闵可夫斯基理论中的“投影体”等物体在对偶理论中也有对偶对应物，即“相交体”。事实上，正是这种“相交体”的双重概念在 Busemann-Petty 问题的最终解决中发挥了关键作用（参见 Gardner [11,12,13]，Zhang [29,30]，Koldobsky [17,18,19,20]，Gardner，Koldobsky 和 Schlumprecht [15]）。加德纳关于几何断层扫描的书 [14] 是经典理论和对偶理论之间相互作用的绝佳参考。闵可夫斯基表明，通过将体积概念与现在称为闵可夫斯基加法的凸体相结合，可以自然地发展出后来被称为布伦-闵可夫斯基理论的东西。 20 世纪 60 年代初，Firey [9] 引入并研究了 Minkowski 加法的 Lp 推广。在 1990 年代的[24, 25]中，这些 Minkowski-Firey Lp 和与体积概念相结合，形成了 Brunn-Minkowski 理论的胚胎 Lp 版本。不难看出，经典勒让德椭球属于对偶 Brunn-Minkowski 理论。这一观察结果让作者提出了一个明显的问题：布伦-闵可夫斯基理论中勒让德椭球的对偶类比是什么？答案是由[26]中引入的新椭球体给出的。这个新的椭球实际上属于 L2-Brunn-Minkowski 理论。 Brunn-Minkowski 理论和对偶 BrunnMinkowski 理论之间的对偶性本质尚不清楚。 Brunn-Minkowski理论的研究对象是凸体，而对偶Brunn-Minkowski理论的研究对象是星体。 Brunn-Minkowski 理论的基本泛函通常表示为涉及支撑函数和曲率函数的积分。对偶 BrunnMinkowski 理论的基本泛函是涉及径向函数的积分。理解 Brunn-Minkowski 理论及其对偶之间的对偶性质的第一步是扩展 Brunn-Minkowski 理论的一些泛函，以便为星体（而不是凸体）定义它们，并为这些泛函提供仅涉及径向函数（而不是支撑函数和曲率函数）的新定义。在本文中，第一步是针对 Brunn-Minkowski 理论的一个对象完成的：Г−2-椭球。信息论的核心问题之一是如何从噪声信号中提取有用信息。设 x0 ∈ R 为传输信号。接收信号的简单模型是随机向量 x ∈ R，R 上的概率分布为 p(x− x0) dx，其中概率测度 p(x) dx 的平均值为 0。假设重复传输相同的信号，并且 x1,…。 。 。 , xN 是接收到的信号。传输信号的最佳估计是什么？该估计的误差是多少？一种可能的估计是平均值 x = x1 + · · ·+ xN N 。根据中心极限定理，随着 N 变大，随机变量 x 的分布接近高斯分布，其平均值为 x0 ，协方差矩阵为 C/ √ N ，其中矩阵 C 由下式给出： Cij = ∫
Associated with each body K in Euclidean n-space R is an ellipsoid Γ2K called the Legendre ellipsoid of K. It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In an earlier paper the authors showed that corresponding to each convex body K ⊂ R is a new ellipsoid Γ−2K that is in some sense dual to the Legendre ellipsoid. The Legendre ellipsoid is an object of the dual Brunn–Minkowski theory, while the new ellipsoid Γ−2K is the corresponding object of the Brunn–Minkowski theory. The present paper has two aims. The first is to show that the domain of Γ−2 can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If K is a star shaped set, then Γ−2K ⊂ Γ2K, with equality if and only if K is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory – the Cramer-Rao inequality. Associated with each body K in Euclidean n-space R is an ellipsoid Γ2K called the Legendre ellipsoid of K. The Legendre ellipsoid is a basic concept from classical mechanics. It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In [26] the authors showed that corresponding to each convex body K ⊂ R is a new ellipsoid Γ−2K. The results in this paper hint at a remarkable duality between this new ellipsoid and the Legendre ellipsoid. The present paper has two aims. The first is to show that the domain of Γ−2 can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If K is a star shaped set, then (1) Γ−2K ⊂ Γ2K, with equality if and only if K is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory – the Cramer-Rao inequality. The Brunn-Minkowski theory (often called the theory of mixed volumes) is the heart of analytic convex geometry. Many of the fundamental ingredients of the theory were developed by Minkowski a century ago. Schneider’s book [28] is the classical reference for the subject. Over the years the tools of the Brunn-Minkowski theory have proven to be remarkably effective in solving inverse problems for which the data involves projections of convex bodies. A quarter of a century ago, the elements of a dual Brunn-Minkowski theory were introduced in [22] (and related papers). The basic idea of the dual theory is to replace the projections 1991 Mathematics Subject Classification. 52A40, 94A17. 1 2 ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG of the Brunn-Minkowski theory with intersections. In [23] it was shown that there is in fact a “dictionary” between the theories. Not only do concepts like the “elementary mixed volumes” of the classical theory become the “elementary dual mixed volumes” of the dual theory, but even objects such as the “projection bodies” of the Brunn-Minkowski theory have dual counterparts, “intersection bodies”, in the dual theory. In fact, it was this dual notion of “intersection body” that played a key role in the ultimate solution of the Busemann-Petty problem (see e.g. Gardner [11, 12, 13], Zhang [29, 30], Koldobsky [17, 18, 19, 20], Gardner, Koldobsky, and Schlumprecht [15]). Gardner’s book on geometric tomography [14] is an excellent reference for the interplay between the classical and dual theories. Minkowski showed that what was to become known as the Brunn-Minkowski theory could be developed naturally by combining the notion of volume with an addition of convex bodies now known as Minkowski addition. In the early 1960’s, Firey [9] introduced and studied an Lp generalization of Minkowski addition. In the 1990’s, in [24, 25], these Minkowski-Firey Lp-sums were combined with the notion of volume to form embryonic Lp versions of the Brunn-Minkowski theory. It is easily seen that the classical Legendre ellipsoid belongs to the dual Brunn-Minkowski theory. This observation led the authors to the obvious question: What is the dual analog of the Legendre ellipsoid in the Brunn-Minkowski theory? The answer was given by the new ellipsoid introduced in [26]. This new ellipsoid actually belongs to the L2-Brunn-Minkowski theory. The nature of the duality between the Brunn-Minkowski theory and the dual BrunnMinkowski theory is not understood. The objects of study of the Brunn-Minkowski theory are convex bodies, while the objects of study of the dual Brunn-Minkowski theory are star bodies. The basic functionals of the Brunn-Minkowski theory are often expressed as integrals involving the support and curvature functions. The basic functionals of the dual BrunnMinkowski theory are integrals involving radial functions. A first step in understanding the nature of the duality between the Brunn-Minkowski theory and its dual is to extend some of the functionals of the Brunn-Minkowski theory so that they are defined for star bodies (rather than convex bodies) and to provide new definitions of these functionals that involve only radial functions (rather than support and curvature functions). In this article, this first step is accomplished for one object of the Brunn-Minkowski theory: the Γ−2-ellipsoid. One of the central problems in information theory is how to extract useful information from noisy signals. Let x0 ∈ R be the transmitted signal. A simple model for the received signal is a random vector x ∈ R with a probability distribution p(x− x0) dx on R, where the probability measure p(x) dx has mean 0. Suppose that the same signal is transmitted repeatedly and that x1, . . . , xN are the received signals. What is the best estimate for the transmitted signal, and what is the error of this estimate? One possible estimate is the mean x = x1 + · · ·+ xN N . By the central limit theorem, as N becomes large, the distribution of the random variable x approaches a Gaussian with mean x0 and covariance matrix C/ √ N , where the matrix C is given by Cij = ∫