The Concentration Of Measure Phenomenon(Ledoux)...

《The Concentration Of Measure Phenomenon(Ledoux)... 》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、THECONCENTRATIONOFMEASUREPHENOMENONMichelLedouxDepartementdeMathematiquesUniversitePaul-Sabatier31062,Toulouse(France)ledoux@cict.frhttp://www.lsp.ups-tlse.fr/Ledoux/Preliminaryversion(September2000)Firstrevision(December2000)12TABLEOFCONTENTSINTRODUCTIONp.51.CONCEN

2、TRATIONFUNCTIONSANDDEVIATIONINEQUALITIESp.111.1Firstexamplesp.111.2Concentrationfunctionsp.141.3Deviationinequalitiesp.171.4Observablediametersp.281.5Expansioncoecientsp.301.6Laplaceboundsandinmum-convolutionsp.31Notesandremarksp.382.ISOPERIMETRICANDFUNCTIONALEXAMPLE

3、Sp.392.1Isoperimetricexamplesp.392.2Brunn-Minkowskiinequalitiesp.512.3Semigrouptoolsp.58Notesandremarksp.663.CONCENTRATIONANDGEOMETRYp.693.1Spectrumandconcentrationp.693.2Spectralanddiameterboundsp.773.3Levyfamiliesp.803.4Topologicalapplicationsp.823.5Euclideansection

4、sofconvexbodiesp.87Notesandremarksp.944.CONCENTRATIONINPRODUCTSPACESp.964.1Martingalemethodsp.9734.2Convexhullapproximationp.1034.3Controlbyseveralpointsp.1124.4Convexinmum-convolutionp.1164.5Theexponentialmeasurep.119Notesandremarksp.1275.ENTROPYANDCONCENTRATIONp.129

5、5.1ConcentrationandlogarithmicSobolevinequalitiesp.1295.2Productmeasuresp.1375.3ModiedlogarithmicSobolevinequalitiesp.1435.4Discretesettingsp.1535.5Covarianceidentitiesp.160Notesandremarksp.1626.TRANSPORTATIONCOSTINEQUALITIESp.1656.1Informationinequalitiesandconcentra

6、tionp.1656.2Quadratictransportationcostinequalitiesp.1696.3Transportationforproductandnon-productmeasuresp.177Notesandremarksp.1847.SHARPBOUNDSONGAUSSIANANDEMPIRICALPROCESSESp.1867.1Gaussianprocessesp.1867.2Boundsonempiricalprocessesp.1947.3Sharperboundsviatheentropicm

7、ethodp.199Notesandremarksp.2098.FURTHERILLUSTRATIONSp.2118.1Concentrationofharmonicmeasuresp.2118.2Concentrationforindependentpermutationsp.2178.3Subsequences,percolation,assignmentp.2228.4Thespinglassfreeenergyp.2268.5Concentrationofrandommatricesp.233Notesandremarksp

8、.235REFERENCESp.237INDEXp.2494INTRODUCTIONTheaimofthisbookistopresentthebasicaspectsofthecon-centrationofmeasurepheno