TheauthorsestablishsomeuniformestimatesforthedistancetohalfwaypointsofminimalgeodesicsintermsofthedistantcetoendpointsonsometypesofRiemannianmanifolds,andthenprovesometheoremsaboutthefinitegenerationoffundamentalgroupofRiemannianmanifoldwithnonnegativeRiccicurvature,whichsupportthefamousMilnorconjecture.