On the magnitudes of compact sets in Euclidean spaces
On the magnitudes of compact sets in Euclidean spaces
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关于欧几里得空间中紧集的大小
DOI：
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
A. Carbery
中科院分区：
文献类型：
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作者：
Juan Antonio Barceló;A. Carbery
度量空间的大小的概念是由Leinster引入的，并在Leinster，Meckes和Willerton的作品中发展，但欧氏空间中熟悉的集合的大小仅在相对较少的情况下被理解。本文研究了欧氏空间中紧集的大小。我们首先描述的渐近的大小，这样的集合在小规模和大规模的制度。然后，我们考虑的幅度紧凸集与非空内部的欧氏空间的奇数维，并将它们的边界行为的解决方案，某些自然相关的高阶椭圆边值问题的外部区域。我们进行计算，导致一个算法的明确评估的幅度的球，这建立了凸的幅度猜想Leinster和Willerton在特殊情况下的球在三维空间。一般来说，奇数维球的大小是其半径的有理函数，从而反驳了莱因斯特-威勒顿猜想的一般形式。除了傅立叶分析和偏微分方程技术，参数还涉及一些组合的考虑。
abstract:The notion of the magnitude of a metric space was introduced by Leinster and developed in works by Leinster, Meckes and Willerton, but the magnitudes of familiar sets in Euclidean space are only understood in relatively few cases. In this paper we study the magnitudes of compact sets in Euclidean spaces. We first describe the asymptotics of the magnitude of such sets in both the small- and large-scale regimes. We then consider the magnitudes of compact convex sets with nonempty interior in Euclidean spaces of odd dimension, and relate them to the boundary behaviour of solutions to certain naturally associated higher order elliptic boundary value problems in exterior domains. We carry out calculations leading to an algorithm for explicit evaluation of the magnitudes of balls, and this establishes the convex magnitude conjecture of Leinster and Willerton in the special case of balls in dimension three. In general the magnitude of an odd-dimensional ball is a rational function of its radius, thus disproving the general form of the Leinster-Willerton conjecture. In addition to Fourier-analytic and PDE techniques, the arguments also involve some combinatorial considerations.
关于欧几里德空间子集的渐近幅
影响因子：
0.5
作者：
Leinster T
通讯作者：
Leinster T