广义Steiner过程的随机收敛性

中文摘要 AI 辅助

众所周知，随机Steiner过程几乎必然收敛到一个欧几里得球。另一方面，如本文所示，平面随机抖动过程在Banach-Mazur距离下几乎必然收敛到一个三角形。为了探究这些现象是普遍存在的还是例外情况，我们研究了称为λ-Steiner对称化的操作的随机迭代，该操作将Steiner对称化和抖动作为特例包含在内。我们建立了此类过程几乎必然收敛和发散的抽象判据，并由此获得了凸几何中许多其他对称化操作的判据。基于椭圆和三角形的联合刻画，我们在平面情形下完整解决了我们的问题。

英文摘要

It is well-known that a random Steiner process converges almost surely to a Euclidean ball. On the other hand, as we show in this article, a planar random shaking process converges almost surely to a triangle (in the Banach-Mazur distance). Asking whether these phenomena are generic or exceptional, we study random iterations of an operation called $λ$-Steiner symmetrization, which includes Steiner symmetrization and shaking as special cases. We establish abstract criteria for almost sure convergence and divergence of such processes, obtaining criteria for many other symmetrization operations from convex geometry as a byproduct. Based on a joint characterization of ellipses and triangles, we give a full resolution to our question in the planar case.

发表机构

• Technion – Israel Institute of Technology（以色列理工学院）
• University of Szeged（塞格德大学）
• HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与教育网络阿尔弗雷德·雷尼数学研究所）

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