Convexity and Concentration PDF
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The IMA Volumes in Mathematics and its Applications Eric Carlen Mokshay Madiman Elisabeth M. Werner Editors Convexity and Concentration The IMA Volumes in Mathematics and its Applications Volume 161 Serieseditor FadilSantosa,UniversityofMinnesota,MN,USA Institute for Mathematics and its Applications (IMA) The Institute for Mathematics and its Applications (IMA) was established in 1982 as a result of a National Science Foundation competition. The mission of theIMAistoconnectscientists,engineers,andmathematiciansinordertoaddress scientific and technological challenges in a collaborative, engaging environment, developing transformative, new mathematics and exploring its applications, while training the next generation of researchers and educators. To this end the IMA organizesawidevarietyofprograms,rangingfromshortintenseworkshopsinareas ofexceptionalinterestandopportunitytoextensivethematicprogramslastingnine months.TheIMAVolumesareusedtodisseminateresultsoftheseprogramstothe broaderscientificcommunity. The full list of IMA books can be found at the Web site of the Institute for MathematicsanditsApplications: http://www.ima.umn.edu/springer/volumes.html. PresentationmaterialsfromtheIMAtalksareavailableat http://www.ima.umn.edu/talks/. Videolibraryisat http://www.ima.umn.edu/videos/. FadilSantosa,DirectoroftheIMA Moreinformationaboutthisseriesathttp://www.springer.com/series/811 Eric Carlen • Mokshay Madiman Elisabeth M. Werner Editors Convexity and Concentration 123 Editors EricCarlen MokshayMadiman DepartmentofMathematics DepartmentofMathematicalSciences RutgersUniversity UniversityofDelaware Piscataway,NJ,USA Newark,DE,USA ElisabethM.Werner DepartmentofMathematics CaseWesternReserveUniversity Cleveland,OH,USA ISSN0940-6573 ISSN2198-3224 (electronic) TheIMAVolumesinMathematicsanditsApplications ISBN978-1-4939-7004-9 ISBN978-1-4939-7005-6 (eBook) DOI10.1007/978-1-4939-7005-6 LibraryofCongressControlNumber:2016961208 MathematicsSubjectClassification(2010):46,52,58,60 ©SpringerScience+BusinessMediaLLC2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerScience+BusinessMediaLLC Theregisteredcompanyaddressis:233SpringStreet,NewYork,NY10013,U.S.A. Foreword ThisvolumeisbasedontheresearchfocusattheIMAduringtheSpringsemesterof 2015.TheAnnualThematicProgramcoveringthisperiodwas“DiscreteStructures: AnalysisandApplications.”TheprogramwasorganizedbySergeyBobkov,Jerrold Griggs,PennyHaxell,MichelLedoux,BennySudakov,andPrasadTetali.Manyof the topics presented in this volume were discussed in the last five workshops that tookplaceduringtheyear.Wethanktheorganizersoftheworkshops,thespeakers, workshop participants, and visitors to the IMA who contributed to the scientific life at the institute and to the successful program. In particular, we thank volume editorsEricCarlen,MokshayMadiman,andespeciallyElisabethWerner,whoalso servedasassociatedirectoroftheIMAduringtheyear.Finally,wearegratefulto theNationalScienceFoundationforitssupportoftheIMA. Minneapolis,MN,USA FadilSantosa v Preface The 2014–2015 Annual Thematic Program at the Institute for Mathematics and its Applications (IMA) was Discrete Structures: Analysis and Applications. The program was organized by Sergey Bobkov (University of Minnesota), Jerrold Griggs (University of South Carolina), Penny Haxell (University of Waterloo), MichelLedoux(PaulSabatierUniversityofToulouse),BennySudakov(University ofCalifornia,LosAngeles),andPrasadTetali(GeorgiaInstituteofTechnology). Convexity and concentration phenomena were the focus during the spring semester of 2015, and this volume presents some of the research topics discussed during this period. We have particularly encouraged authors to write surveys of research problems, thus making state-of-the-art results more conveniently and widely available. The volume addresses the themes of five workshops held during thespringsemesterof2015: • Convexity and Optimization: Theory and Applications, held February 23–27, 2015, at IMA and organized by Nina Balcan (Carnegie-Mellon University), HenrikChristensen(GeorgiaInstituteofTechnology),WilliamCook(University of Waterloo), Satoru Iwata (University of Tokyo), and Prasad Tetali (Georgia InstituteofTechnology) • ThePowerofRandomnessinComputation,heldMarch16–20,2015,atGeorgia Institute of Technology and organized by Dana Randall (Georgia Institute of Technology),PrasadTetali(GeorgiaInstituteofTechnology),SantoshVempala (GeorgiaInstituteofTechnology),andEricVigoda(KoreaAdvancedInstituteof ScienceandTechnology(KAIST)) • Information Theory and Concentration Phenomena, held April 13–17, 2015, at IMA and organized by Sergey Bobkov (University of Minnesota, Twin Cities), Michel Ledoux (Université de Toulouse III (Paul Sabatier)), and Joel Tropp (CaliforniaInstituteofTechnology) • AnalyticToolsinProbabilityandApplications,heldApril27–May01,2015,at IMA and organized by Sergey Bobkov (University of Minnesota, Twin Cities), SergeiKislyakov(RussianAcademyofSciences),MichelLedoux(Universitéde ToulouseIII(PaulSabatier)),andAndreiZaitsev(RussianAcademyofSciences) vii viii Preface • GraphicalModels,StatisticalInference,andAlgorithms(GRAMSIA),heldMay 18–22, 2015, at IMA and organized by David Gamarnik (Massachusetts Insti- tute of Technology), Andrea Montanari (Stanford University), Devavrat Shah (Massachusetts Institute of Technology), Prasad Tetali (Georgia Institute of Technology), Rüdiger Urbanke (École Polytechnique Fédérale de Lausanne (EPFL)),andMartinWainwright(UniversityofCalifornia,Berkeley) Discrete Structures: Analysis and Applications attracted intense interest from themathematicalsciencecommunityatlarge.Eachofthefiveworkshopsdrewup to or more than 100 visitors. Aside from the workshops, an annual thematic year at the IMA provided an ideal environment for collaborative work. This program drewamixofexpertsandjuniorresearchersinvariousaspectsofconvexgeometry and probability together with numerous people who apply these areas to other fields. This volume reflects many aspects of the semester, with chapters drawn fromworkshoptalks,annualprogramseminars,andtheresearchinterestsofmany visitors. The volume is organized into two parts. While the classification is of course arbitrarytosomeextentgiventhefluidboundariesbetweenprobabilityandanalysis, Part I: Probability and Concentration contains those contributions that focus primarily on problems motivated by probability theory, while Part II: Convexity andConcentrationforSetsandFunctionscontainsthosecontributionsthatfocus primarilyonproblemsmotivatedbyconvexgeometryandgeometricanalysis. Acknowledgments Nosinglevolumecouldpossiblycoveralltheactiveandimportantareasof researchinconvexity,probability,andrelatedfieldsthatwerepresentedattheIMA,andwemake noclaimofcomprehensiveness.However,wethinkthatthisvolumepresentsareasonableselection ofinterestingareas,writtenbyleadingexpertswhohavesurveyedthecurrentstateofknowledge andposedconjecturesandopenquestionstostimulatefurtherresearch.Wethanktheauthorsfor their generous donation of time and expertise. Needless to say that without them, this volume wouldnothavebeenpossible. WethankA.Beveridge,J.R.Griggs,L.Hogben,G.Musiker,andP.Tetali,theeditorsofthe 2014–2015fallsemestervolumeRecentTrendsinCombinatorics,fortheiradviceandtheirmany helpfulcomments. WethanktheIMAandtheirstaffforwonderfullystimulatingandproductivelong-termvisits. WebelievethattheIMAisacriticalnationalresourceformathematics.TheDiscreteStructures: AnalysisandApplicationsprogramwillhavealastingimpactonresearchinconvexity,probability, and related fields, and we hope this volume will enhance that impact. We are grateful for the opportunitytobepartofit. Piscataway,NJ,USA EricCarlen Newark,DE,USA MokshayMadiman Cleveland,OH,USA ElisabethM.Werner Contents PartI ProbabilityandConcentration InterpolationofProbabilityMeasuresonGraphs............................ 3 ErwanHillion EntropyandThinningofDiscreteRandomVariables....................... 33 OliverJohnson ConcentrationofMeasurePrincipleandEntropy-Inequalities............. 55 Paul-MarieSamson StructuredRandomMatrices................................................... 107 RamonvanHandel RatesofConvergenceforEmpiricalSpectralMeasures:ASoft Approach.......................................................................... 157 ElizabethS.MeckesandMarkW.Meckes Concentration of Measure Without Independence: A Unified ApproachViatheMartingaleMethod......................................... 183 AryehKontorovichandMaximRaginsky StrongData-ProcessingInequalitiesforChannelsandBayesian Networks.......................................................................... 211 YuryPolyanskiyandYihongWu AnApplicationofaFunctionalInequalitytoQuasi-Invariancein InfiniteDimensions .............................................................. 251 MariaGordina Borell’sFormulaonaRiemannianManifoldandApplications ............ 267 JosephLehec FourthMomentsandProducts:UnifiedEstimates........................... 285 IvanNourdinandGiovanniPeccati ix x Contents AsymptoticExpansionsforProductsofCharacteristicFunctions UnderMomentAssumptionsofNon-integerOrders ........................ 297 SergeyG.Bobkov PartII ConvexityandConcentrationforSetsandFunctions Non-standard Constructions in Convex Geometry: Geometric MeansofConvexBodies......................................................... 361 VitaliMilmanandLiranRotem RandomizedIsoperimetricInequalities ....................................... 391 GrigorisPaourisandPeterPivovarov ForwardandReverseEntropyPowerInequalitiesinConvexGeometry.. 427 MokshayMadiman,JamesMelbourne,andPengXu Log-ConcaveFunctions.......................................................... 487 AndreaColesanti OnSomeProblemsConcerningLog-ConcaveRandomVectors............ 525 RafałLatała Stability Results for Some Geometric Inequalities and Their FunctionalVersions.............................................................. 541 UmutCaglarandElisabethM.Werner MeasuresofSectionsofConvexBodies........................................ 565 AlexanderKoldobsky On Isoperimetric Functions of Probability Measures Having Log-ConcaveDensitieswithRespecttotheStandardNormalLaw........ 577 SergeyG.Bobkov CountingIntegerPointsinHigher-DimensionalPolytopes ................. 585 AlexanderBarvinok TheChainRuleOperatorEquationforPolynomials andEntireFunctions ............................................................ 613 HermannKönigandVitaliMilman
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