[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82105-en":3,"doc-seo-82105-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82105,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Online Komlós Converges to Mean Curvature Flow","Online Komlós game studies online vector balancing through a zero-sum interaction where Paul adaptively selects n vectors each round (bounded in the Euclidean unit ball) and Carol chooses signs to update the state vector y in R^m. After T rounds, Paul maximizes the ℓ∞ norm of y while Carol minimizes it. As T grows, the game value has leading order pT/2τ, with τ the mean-curvature-flow extinction time of the unit cube in R^m; for n≥m−1 this flow is the mean curvature flow, yielding 1/√(2τ)=Θ(√log m). The work connects deterministic games, mean curvature flow, and Banaszczyk’s ℓ2 Beck–Fiala analogue.","arXiv :2607 .08943v1 [math .CO] 9 Jul 2026  \nONLINE KOML´OS CONVERGES TO MEAN CURVATURE FLOW  \nNESTOR GUILLEN AND VLADIMIR A. KOBZAR  \nDedicated to Robert V. Kohn, 1953-2026  \nAbstract. 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 Koml´os game, two players, Paul and Carol, update the state vector y in Rm , 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 ℓ∞ 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 pT/2τ, where τ is the extinction time of the unit cube in Rm under a curvature-based flow characterized by the values of m and n. When n ≥ m − 1, this flow is the mean curvature flow, and we show that 1/ √ 2τ = Θ( √log m) . Our results build upon the work of Kohnand Serfaty on deterministic games and mean curvature flow, combined with Banaszczyk’s ℓ2 analogue of the Beck-Fiala theorem. As the large T limit of the online Koml´os game amounts to a localization of the classic Koml´os problem, we hope this work can shed light on this and other vector balancing problems. Our results generalize to the version of the online Koml´os game with the final value given by an arbitrary normin Rm .  \n1. Introduction  \nDiscrepancy theory covers a wide array of problems and methods in combinatorics, geometry and analysis that involve approximation of a mathematical object by discrete elements [89, 111] . This field has been a long-standing focus of computer science and scientific computing, ranging from differential privacy and analog-to-digital conversion to causal inference and learning theory [8, 38 , 40 , 64 , 67 , 92] . One of the main problems in combinatorial discrepancy is vector balancing: given vectors a 1 , . . . , ak in a normed space, our objective is to choose signs ε 1 , . . . ,εk ∈ {−1, 1} in order to approximate the zero vector, i.e., minimize  \n∥ε1 a 1 + ... + εkak ∥ .  \nWe will focus on a version of this problem posed as a zero-sum game between two players, Paul and Carol.1 The game is played for T rounds where Paul reveals a set of k = nT vectors in Rm for m ≥ 2 to Carol not all at once, but sequentially in T batches of size n.  \nSince the case when T = 1 corresponds to the classic open problem known as the Koml´os conjecture, we will refer to our game as the online Koml´os game, formally defined as follows. At round t = 0, we initialize the vector yt ∈ Rm with y0 = 0 . In each subsequent round from t = 1 until the final round T, Paul chooses n vectors a 1,t , . . . , an,t ∈ Rm , each with at most unit Euclidean length, and reveals them to Carol. Then Carol irrevocably chooses n signs ε¯t = (ε1,t,...,εn,t) ∈ {−1, 1}n , and we update  \nyt = yt−1 + (ε1,ta 1,t + ... + εn,tan,t) .  \nWe will write this update more compactly as yt = yt−1 + At ε¯t where At denotes the m × n matrix whose columns are given by the vectors a 1,t , . . . , an,t. Paul’s objective is to make the norm ∥yT ∥∞ as large as possible, while Carol’s objective is to make it as small as possible.  \nPaul is referred to as an adaptive adversary because he can choose each At based on Carol’s choices of ε¯τ ’s at earlier times τ \u003C t. Accordingly, the value KT (m, n) of the online Koml´os game is given by  \nKT (m, n) := a1x mε¯i1n ... x mε¯ ∥A1 ε¯1 + ... + AT ε¯T ∥∞ (1.1)  \nDate: July 13, 2026 .  \n1These names are mnemonics for pusher of vectors and chooser of signs, respectively, in the games introduced by Spencer.  \n2 NESTOR GUILLEN AND VLADIMIR A. KOBZAR  \nwhere the feasible set of each At is given by all ","cbCaieWFI7wLTvWf","https://ap.wps.com/l/cbCaieWFI7wLTvWf","pdf",593070,3,1,38,"English","en",105,"# Introduction\n## Online Komlós game definition\n## Relation to mean curvature flow and continuum limits\n## Main asymptotic result","[{\"question\":\"What is the online Komlós game?\",\"answer\":\"Paul and Carol play a sequential zero-sum game across T rounds. Paul chooses n vectors in the Euclidean unit ball each round, Carol assigns signs, and the state vector y is updated by adding the signed batch; the game ends by evaluating the ℓ∞ norm of y.\"},{\"question\":\"How do the players' objectives differ?\",\"answer\":\"Paul aims to maximize the final ℓ∞ norm of the state vector y, while Carol aims to minimize that same norm throughout the game.\"},{\"question\":\"What is the asymptotic behavior of the game value as T becomes large?\",\"answer\":\"As T→∞, the leading-order term of the game value is pT/2τ, where τ is the extinction time of the unit cube under a curvature-based flow. When n≥m−1, this curvature-based flow becomes the mean curvature flow and satisfies 1/√(2τ)=Θ(√log m).\"}]",1784178232,96,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"online-komlos-converges-to-mean-curvature-flow","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/online-komlos-converges-to-mean-curvature-flow/82105/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-20","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the online Komlós game?","Question",{"text":75,"@type":76},"Paul and Carol play a sequential zero-sum game across T rounds. Paul chooses n vectors in the Euclidean unit ball each round, Carol assigns signs, and the state vector y is updated by adding the signed batch; the game ends by evaluating the ℓ∞ norm of y.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the players' objectives differ?",{"text":80,"@type":76},"Paul aims to maximize the final ℓ∞ norm of the state vector y, while Carol aims to minimize that same norm throughout the game.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the asymptotic behavior of the game value as T becomes large?",{"text":84,"@type":76},"As T→∞, the leading-order term of the game value is pT/2τ, where τ is the extinction time of the unit cube under a curvature-based flow. When n≥m−1, this curvature-based flow becomes the mean curvature flow and satisfies 1/√(2τ)=Θ(√log m).","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]