| This paper investigates some geometric inequalities and applications for dif-ferent measures.Based on the research subject,it is composed of the following parts.Given a measureμon R~n,Chapter 2 extends L_p-mixed volumes and L_p-surface area measures to L_p-mixedμ-measures and L_p-surfaceμ-area measures,respectively.The integral representation for the L_p-surfaceμ-area measure will be given.Using this integral representation,some geometric inequalities are proved,including L_p-Minkowski and L_p-Brunn-Minkowski inequalities for measures.Moreover,the existence property is proved for the solution of the related L_p-Minkowski problem for measures.Chapter 3 discusses the related L_p-Minkowski problems for measures.The existence theorem will be shown and was used to obtain a new notion,which is an extension of the L_p-Blaschke addition.Some geometric inequalities about L_p-Blaschke addition are obtained.Lutwak-Yang-Zhang established the Orlicz centroid inequality for convex body,and conjectured it can be extend to all star bodies.Zhu proved this in-equality for star bodies,and proved the equality condition when?is strictly convex.Without this assumption that?is strictly convex,Chapter 4 will discuss this equality condition.Volume difference inequalities are designed to estimate the volume difference of two bodies in terms of the maximal or minimal of the volume difference of sec-tions of these bodies section.In the last chapter,two such inequalities established in[49]and[30]are extended from the hyperplane to the case of sections of arbi-trary dimensions.Furthermore,these two inequalities are extended to measure difference inequalities,for measure with 1/τconcave density,whereτ>0 is an integer. |
