在线Komlós收敛到平均曲率流
Online Komlós converges to mean curvature flow
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中文总结 AI 辅助
研究受组合偏差理论启发的在线Komlós博弈，保罗和卡罗尔更新状态向量\(y\)，目标分别是最大化和最小化\(y\)的\(\ell_\infty\)范数。通过特定方法确定博弈值主导项，推广到任意范数情形，为向量平衡问题提供启示。
中文摘要 AI 辅助
我们确定了一个受组合偏差理论中经典向量平衡问题启发的博弈的渐近性质。在这个我们称为在线Komlós博弈的游戏中，保罗和卡罗尔两个玩家在\(\mathbb{R}^m\)中更新初始位于\(0\)的状态向量\(y\)。每一轮，保罗在欧几里得单位球中自由选择一组\(n\)个向量，卡罗尔为每个向量选择是否保持不变或改变其符号。然后将得到的向量都加到\(y\)上，游戏进入新一轮。\(T\)轮后游戏结束，确定状态向量\(y\)的\(\ell_\infty\)范数。保罗的目标是最大化这个范数，卡罗尔的目标是最小化它。当\(T\)变大时，我们确定这个博弈值的主导项是\(\sqrt{T/2\tau}\)，其中\(\tau\)是\(\mathbb{R}^m\)中单位立方体在基于\(m\)和\(n\)值的曲率流作用下的灭绝时间。当\(n\geq m - 1\)时，这个流是平均曲率流，且我们证明\(1/\sqrt{2\tau}=\Theta(\sqrt{\log m})\)。我们的结果基于Kohn和Serfaty关于确定性博弈和平均曲率流的工作，结合了Banaszczyk的Beck - Fiala定理的\(\ell^2\)类似物。由于在线Komlós博弈的大\(T\)极限相当于经典Komlós问题的局部化，我们希望这项工作能为这个及其他向量平衡问题提供启示。我们的结果推广到了在线Komlós博弈的版本，其最终值由\(\mathbb{R}^m\)中的任意范数给出。
英文摘要
We determine the asymptotics of a game inspired by classic vector balancing problems in combinatorial discrepancy theory. In this game, which we call the online Komlós game, two players, Paul and Carol, update the state vector $y$ in $\mathbb{R}^m$, initially placed at $0$. At each round, Paul chooses freely a set of $n$ vectors in the Euclidean unit ball, and Carol chooses, for each such vector, whether to leave it unchanged or reverse its sign. The resulting vectors are all added to $y$, and the game proceeds to a new round. After $T$ rounds, the game ends, and the $\ell_\infty$ norm of the state vector $y$ is determined. Paul's objective throughout the game is to maximize this norm, and Carol's objective is to minimize it. As $T$ gets large, we establish that the leading order term of the value of this game is $\sqrt{T/2τ}$, where $τ$ is the extinction time of the unit cube in $\mathbb{R}^m$ under a curvature-based flow characterized by the values of $m$ and $n$. When $n\geq m-1$, this flow is the mean curvature flow, and we show that $1/\sqrt{2τ} =Θ(\sqrt{\log m})$. Our results build upon the work of Kohn and Serfaty on deterministic games and mean curvature flow, combined with Banaszczyk's $\ell^2$ analogue of the Beck-Fiala theorem. As the large $T$ limit of the online Komlós game amounts to a localization of the classic Komlós problem, we hope this work can shed light on this and other vector balancing problems. Our results generalize to the version of the online Komlós game with the final value given by an arbitrary norm in $\mathbb{R}^m$.
