Table Of ContentThe 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
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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
