The Spherical Harmonics In Convex Geometric Analysis And Related Problems

The researches of this thesis belongs to the theory of convex geometric anal-ysis and spherical harmonic analysis, and devoted to the study of spherical har-monics in convex geometric analysis and related problems. Spherical integraltransforms are important tools in modern geometry and have a wide rang of ap-plications in functional analysis, geometric tomography, convex geometry analysisand statistical geometry (see, e.g.,[46,62,80,142]). The thesis originates from theuse of harmonics analysis in convex geometry analysis, and even more precisely,the applications of spherical integral transforms to the questions of uniquenessand stability for convex bodies and the questions of volume inequalities. Theseproblems have attracted increased interest for this direction and some importantresult have been obtained.In chapter2, we study the determination of star bodies. Centroid bodieswere first defined and investigated by Petty [124], but the concept had previouslyappeared in work of Dupin (see e.g.[139], Section7.4, for references). In1990,Lutwak [95] generalize the concept of a centroid body and introduce p-centroidbody. In this chapter, we prove that an origin-symmetric star body is uniquelydetermined by its p-centroid body. Furthermore, using spherical harmonics, weestablish a result for non-symmetric star bodies. As application, we show thatthere is a unique member of ΓpK characterized by having larger volume thanany other member, for all real p≥1that are not even natural numbers, whereΓpK denotes the p-centroid equivalence class of the star body K.In2007, Fleury, Gu′edon and Paouris [41] proved a type of stability result forp-centroid bodies with respect to the geometric distance. In chapter3, we provea stability result for p-centroid bodies with respect to the Hausdorf distance dif-ferent from the stability result of Fleury, Gu′edon and Paouris. As its application, we show that the symmetric convex body is determined by its p-centroid body.In chapter4, we introduce one useful spherical integral transform, p-sinetransform of isotropic measures. For p≥1, sharp isoperimetric inequalities forp-sine transform of isotropic measures and the corresponding reverse inequalitiesare established in this chapter. As applications of our main results, we alsopresent volume inequalities for convex bodies which are in isotropic position.In chapter5, we establish the Gaussian width inequalities for symmetricWulf shapes by a direct approach. We also yield the dual inequality along withthe equality conditions. These new inequalities have as special cases previouslyobtained Barthe’s mean width inequalities for even isotropic measures and itsdual form.In chapter6, we introduce an Orlicz zonoid operator Zφ:(K~n)~2â†'Knwhichhas the mean zonoid operator as special case and show that the operator whichactually maps (K~n)~2to Kn0is commutativity, homogeneity of degree1, GL(n)covariance and continuous. We also establish the afne inequality for Orliczzonoid operator.