Analytic Aspects of Convexity PDF
Preview Analytic Aspects of Convexity
Springer INdAM Series 25 Gabriele Bianchi Andrea Colesanti Paolo Gronchi Editors Analytic Aspects of Convexity Springer INdAM Series Volume 25 Editor-in-Chief G.Patrizio SeriesEditors C.Canuto G.Coletti G.Gentili A.Malchiodi P.Marcellini E.Mezzetti G.Moscariello T.Ruggeri Moreinformationaboutthisseriesathttp://www.springer.com/series/10283 Gabriele Bianchi • Andrea Colesanti (cid:129) Paolo Gronchi Editors Analytic Aspects of Convexity 123 Editors GabrieleBianchi AndreaColesanti Dept.ofMathematicsandComputer Dept.ofMathematicsandComputer Science“U.Dini” Science“U.Dini” UniversityofFlorence UniversityofFlorence Florence,Italy Florence,Italy PaoloGronchi Dept.ofMathematicsandComputer Science“U.Dini” UniversityofFlorence Florence,Italy ISSN2281-518X ISSN2281-5198 (electronic) SpringerINdAMSeries ISBN978-3-319-71833-0 ISBN978-3-319-71834-7 (eBook) https://doi.org/10.1007/978-3-319-71834-7 LibraryofCongressControlNumber:2018934461 ©SpringerInternationalPublishingAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This volume contains a selection of papers presented at the INdAM Workshop “Analytic Aspects of Convexity” and closely related works by some of the par- ticipants.Thisconference,heldinRomefromOctober10to14,2016,represented acontinuationofaseriesofconferences,withthesametitle,organizedbyStefano Campi, Rolf Schneider and Aljoša Volcˇicˇ. The previousconferenceswere held in Cortonaevery4yearsfrom1995to2011. Convexity is a notion which plays a significant role in numerous areas of mathematicsanditsusedatesbackseveralcenturies.Itwasessentiallythegeniusof HermannMinkowskiwhich,aroundtheturnofthetwentiethcentury,wasresponsi- blefortheinventionordiscoveryofthatbranchofmathematicsthatnowgoesunder thenameofConvexGeometry.Sincethenmanymathematicianshaveexploredand expanded the field using techniques from differential geometry, measure theory, combinatorics, algebraic geometry, Fourier analysis and probability. Interesting relations to other topics have also been disclosed, and, stealing a sentence from P.M. Gruber’s History of Convexity (in: “Handbook of Convex Geometry”, P.M. GruberandJ.M. Wills, editors,North-HollandPublishingCo., 1993),“partof the fascination of convexity is due to these interrelations”. Nowadays, the research in Convex Geometry has many different souls and purposes. As an example we can refer to the bookswritten by four participantsin the conference:P.M. Gruber (“ConvexandDiscreteGeometry”,Springer,2007),R.Schneider(“ConvexBodies: theBrunn-MinkowskiTheory”,CambridgeUniversityPress,2014),A.Koldobsky (“Fourier Analysis in Convex Geometry”,American MathematicalSociety, 2005) andR.J.Gardner(“GeometricTomography”,CambridgeUniversityPress,2014). The focus of the workshop was on some aspects of convexity where the connection with analysis, either in the methods or in the problems themselves, is particularlystrong.The selectionofpapersin thisvolumeoffersa goodreflection ofthetopicsmostrepresentedinthemeeting. Geometricinequalitiesandproblemsofisoperimetrictypearerepresentedbythe papersbyM.A.HernándezCifreandD.Alonso-Gutiérrez,A.E.Litvak,A.Stancu andJ.YepesNicolás.Thefirstofthesepapersdealswithestimationofintegralsof powersofthemeancurvaturesofthesurfaceofaconvexbodyintermsofgeometric v vi Preface quantities of the body itself, like surface, volume or, more generally, intrinsic volumes. Litvak reports on an old conjecture regarding the simplex of maximum mean width among those inscribed in the unit ball, and the importance of this problemin probabilitytheoryandinformationtheory.Stancu dealswith problems inaffinegeometryandproposesadefinitionofcentro-affinecurvatureandofaffine length for polygons, with the latter satisfying isoperimetric type inequalities. The paper by J. Yepes Nicolás deals with the Brunn-Minkowskiinequality, of central importanceinconvexityandinanalysis. Fourier analysis and geometric tomography are represented here by the paper by A. Koldobsky and D. Wu, which gives estimates on the difference in volume betweentwostarbodiesintermsofmaximalorminimaldifferencebetweenareas ofsectionsofthosebodies.Theseestimatesarecloselyrelatedtofamousproblems in geometric tomography, such as the slicing problem and the Busemann-Petty problem.Fourieranalysisplaysanessentialroleintheproofsoftheestimates. Integralgeometryandthetheoryofvaluationsarerepresentedbythepapersby A.Bernig,J.H.G.FuandG.SolanesandbyD.HugandJ.A.Weis.Thefirstpaper dealswithkinematicformulasandvaluationsoncomplexspaceforms,whileHug andWeis focuson integralgeometryoftensor-valuedgeneralisationsofcurvature measuresofconvexbodies. OurthanksgototheIstitutoNazionalediAltaMatematica“FrancescoSeveri”, whichmadeitpossibletoholdtheworkshopthroughtheirfinancialsupportandby hostingusintheirheadquartersinRome. WehadthehonourofhavingPeterM.Gruber,whosadlypassedawayonMarch 7,2017,asaparticipantandspeakerinthemeeting.Hewasaleadingexpertinthe field and contributeddeeplyto the formationof this researchcommunity.We will allmisshimgreatly. Florence,Italy GabrieleBianchi October2017 AndreaColesanti PaoloGronchi Contents DualCurvatureMeasuresinHermitianIntegralGeometry................ 1 AndreasBernig,JosephH.G.Fu,andGilSolanes EstimatesfortheIntegralsofPoweredi-thMeanCurvatures.............. 19 MaríaA.HernándezCifreandDavidAlonso-Gutiérrez Crofton Formulae for Tensorial Curvature Measures: TheGeneralCase ................................................................ 39 DanielHugandJanA.Weis ExtensionsofReverseVolumeDifferenceInequalities....................... 61 AlexanderKoldobskyandDenghuiWu AroundtheSimplexMeanWidthConjecture................................ 73 AlexanderE.Litvak DiscreteCentro-AffineCurvatureforConvexPolygons..................... 85 AlinaStancu CharacterizingtheVolumeviaaBrunn-MinkowskiTypeInequality ..... 103 JesúsYepesNicolás vii About the Editors Gabriele Bianchi, Andrea Colesanti and Paolo Gronchi are professors at the Departmentof Mathematicsand ComputerScience of the University of Florence. TheirmainresearchinterestisintheanalyticaspectsofConvexGeometry. ix Dual Curvature Measures in Hermitian Integral Geometry AndreasBernig,JosephH.G.Fu,andGilSolanes Abstract The local kinematic formulas on complex space forms induce the structureofacommutativealgebraonthespaceCurvU.n/(cid:2)ofdualunitarilyinvariant curvature measures. Building on the recent results from integral geometry in complexspaceforms,wedescribethisalgebrastructureexplicitlyasapolynomial algebra. This is a short way to encode all local kinematic formulas. We then characterizethe invariantvaluationson complexspace formsleaving the space of invariantangularcurvaturemeasuresfixed. 1 Introduction Let CPn denote the complex space form of holomorphic sectional curvature 4(cid:2) (cid:2) andG(cid:2) itsholomorphicisometrygroup.ByC.CPn(cid:2)/G(cid:2) wedenotethespaceofG(cid:2)- invariantsmoothcurvaturemeasuresonCPn (seebelowforthedefinition). (cid:2) ThespaceC.CPn/G(cid:2) isfinite-dimensional,andseveralgeometricallymeaningful (cid:2) bases were used in [3]. Let ˆ1;:::;ˆm be such a basis. Then there are local kinematicformulas(cf.[4]or[7]) Z X ˆj.P1\gP2;ˇ1\gˇ2/dgD ckj;lˆk.P1;ˇ1/ˆl.P2;ˇ2/: G(cid:2) k;l A.Bernig((cid:2)) InstitutfürMathematik,Goethe-UniversitätFrankfurt,Frankfurt,Germany e-mail:[email protected] J.H.G.Fu DepartmentofMathematics,UniversityofGeorgia,Athens,GA,USA e-mail:[email protected] G.Solanes DepartamentdeMatemàtiques,UniversitatAutònomadeBarcelona,Bellaterra,Spain e-mail:[email protected] ©SpringerInternationalPublishingAG2018 1 G.Bianchietal.(eds.),AnalyticAspectsofConvexity,SpringerINdAMSeries25, https://doi.org/10.1007/978-3-319-71834-7_1
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