Even valuations on convex bodies
Even valuations on convex bodies
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DOI：
10.1090/s0002-9947-99-02240-0
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发表时间：
1999-05
影响因子：
1.3
通讯作者：
Daniel A. Klain
中科院分区：
文献类型：
--
作者：
Daniel A. Klain
eveiL估值的概念是作为体积在欧氏空间的紧致凸子集上的自然推广引入的。最近关于体积的一个表征定理引出了紧凸集上的偶值与格拉斯曼上的连续函数之间的联系。这种联系可以部分地用对称紧凸集的生成分布来描述。我们还探讨了这些表征结果在凸几何和积分几何中的一些结果。最近人们对紧凸集的体积评价的兴趣源于希尔伯特第三问题，它实际上是一个用现代术语重新定义的古老问题。希尔伯特问两个多面体P和Q是否可以被切成有限个P块，…， Pn和Q，…， Qm，通过刚性欧几里得运动，每个Pi与Qi相等，假设P和Q具有相同的体积。Max Dehn[2,4,29]给出了否定的答案，他发现多面体上的泛函在刚性运动的解剖下是不变的，而在等体积多面体上的值是不同的。换句话说，Dehn不变量是不等于体积（在任何归一化下）的多面体上的“简单刚性运动不变量估值”。Dehn的解决方案留下了一个问题，即究竟在什么条件下P和Q意味着刚性运动群的等切分性，尽管这个问题是由Hadwiger在只允许平移（不允许旋转和反射）的情况下解决的（见[2,16,17,26,29]）。在研究这一问题及相关问题的过程中，Hadwiger发现了欧几里得体积在紧凸集上的连续刚体运动不变简单值的表征，即在小于满维的凸集上消失的连续刚体运动不变值。这一结果反过来又导致了1R'中紧凸集上的所有连续刚性运动不变值的完整表征，这些值由内征体积（或Quermassintegrals）[16]张成的实数（n 4）维向量空间组成（也[20,21,31]）。由于许多标准泛函数和积分算子可以被解释为不变值（如内在体积、平均投影、克罗夫顿和运动学公式），因此后来被称为哈德维格刻画定理的东西被证明是一个有价值的工具，可以快速而轻松地证明积分几何中的许多公式和方程。不幸的是，哈德维格最初的证明既长又难[161]。在为哈德维格的卷特征寻找较短的证明时，作者发现了1996年6月24日和1997年9月29日修订后的《Received》。1991数学学科分类。Primary 52A22, 52A38, 52A39, 52B45。部分研究由NSF资助，MSRI资助#DMS 9022140，作者资助#DMS 9626688。？ 该内容从207.46.13.52星期一，2016年10月24日04:15:57 UTC下载
The notion of eveiL valuation is introduced as a natural generalization of volume on cornpact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry. Recent interest in volume as a valuation on compact conlvex sets stems from Hilbert's Third Problem, which is actually an ancient problem recast in modern terms. Hilbert asked if two polytopes P and Q can be each cut into a finite number of pieces P.,... , Pn and Q ,... , Qm with each Pi congruent to Qi by a rigid Euclidean motion, provided that P and Q have the same volume [18]. This question was answered in the negative by Max Dehn [2, 4, 29], who found a functionlal on polytopes that is invariant under dissections over rigid motions, while varying in value among polytopes of equal volume. In other words, the Dehn invariant is a "simple rigid motion invariant valuation" on polytopes that is not equal to volume (under any normalization). Dehn's solution left open the question of exactly what conditions oil P and Q imply equidissectability over the group of rigid motions, although this problem was solved by Hadwiger in the case where only translations (and no rotations nor reflections) are permitted (see [2, 16, 17, 26, 29]). In the course of studying this and related problems, Hadwiger discovered a characterization of Euclidean volume as a continuous rigid motion invariant simple valuation on compact convex sets, that is, a continuous rigid motion invariant valuation that vanishes on convex sets of less than full dimension. This result led in turn to a complete characterization of all continuous rigid motion inivariant valuations on compact convex sets in 1R' as consisting of a real (n 4)-dimensional vector space spanned by the intrinsic volumes (or Quermassintegrals) [16] (also [20, 21, 31]). Since many standard functionals and integral operators can be interpreted as invariant valuations (such as intrinsic volumes, mean projections, Crofton and kinematic formulas), what came to be known as Hadwiger's chtracterization theorem proved to be a valuable tool for generating quick and effortless proofs of many formulas and equations in integral geometry. Unfortunately Hadwiger's original proof was long and difficult [161 While seeking a shorter proof of Hadwiger's volume characterization, the author discovered Received by the editors June 24, 1996 and, in revised form, September 29, 1997. 1991 Mathematics Subject Classification. Primary 52A22, 52A38, 52A39, 52B45. Research supported in part by NSF grants #DMS 9022140 to MSRI and #DMS 9626688 to the author. ? 1999 American Mathematical Society 71 This content downloaded from 207.46.13.52 on Mon, 24 Oct 2016 04:15:57 UTC All use subject to http://about.jstor.org/terms