高斯卷积、内能与Kneser--Poulsen猜想
Gaussian Convolution, Internal Energies, and the Kneser--Poulsen Conjecture
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中文总结 AI 辅助
本文通过定义基于压力律迭代非负性的内能层级，结合高斯边缘化与连续收缩同伦，证明了热流下内能比较的持久性，并由此推出Kneser--Poulsen猜想的多个已知情形，进而猜想该持久性在所有维度对所有凸能量密度成立。
中文摘要 AI 辅助
我们研究了由$1$-Lipschitz映射诱导的内能比较在热流下的持久性。我们基于相关压力律的迭代非负性定义了一个新的内能层级，并证明高斯边缘化导致层级上升。该机制与连续收缩同伦的更强无维数结果相结合，使我们能够证明持久性在层级中依赖于环境维数的各层成立，包括二维情形下的所有凸能量密度。随后我们将这些结果与几何学联系起来，特别是与管道的体积相联系，表明Kneser--Poulsen猜想的几个主要已知情形可由我们的结果推出。部分受此联系启发，我们猜想在所有维度中持久性对所有凸能量密度均成立。我们还讨论了该框架中自然出现的各向同性高斯分布的一个刻画。
英文摘要
We study to what extent the majorisation order between a probability measure and its $1$-Lipschitz image is preserved when both measures undergo Gaussian convolution. We show that the majorisation order is fully preserved in dimensions $n\leq 2$, and obtain dimension-dependent partial preservation in higher dimensions. The perspective taken is that majorisation between densities amounts to comparison through internal energies satisfying a certain pressure condition. Accordingly we introduce a notion of iterated nonnegativity of pressure, closely related to iterated pressures arising in optimal transport, placing the majorisation order within a graded hierarchy of internal energy comparisons. Using the observation that the volume of Euclidean neighbourhoods can be detected from internal energy measurements along the heat flow, we show that our results imply several principal known cases of the Kneser--Poulsen conjecture.
发表机构
- Technion – Israel Institute of Technology（以色列理工学院）
- Shaanxi Normal University（陕西师范大学）
机构由 AI 辅助整理，请以论文原文为准。
