Stability Of Volume Comparison Problems For Intersections And Projections Of Convex Bodies

The topic of this thesis belongs to the extremal theory of convex bodies whichfocuses on intersection of star body and projection of convex body. The thesis devotesto the stability of some inequalities involve intersection and projection. Using Fouriertransform and Radon transform, we obtain several stability results.Our first main goal is to consider the stability problem of the volume comparisonproblems. We provide a new stability in terms of volume quotient instead of volumedifference of Koldobsky’s. In view of the closed relationship between i-Radontransform and i-intersection bodies, applying the method of functional analysis, we getthe stability results of the generalized Funk’s intersection theorem. These results providea new vision for the further study of some important open problems such as the slicingproblem.Next, we establish the stability result of generalized Busemann-Ptty problem withvolume quotient. Specially, we investigate the connections between the new stability inthe volume quotient and stability in the volume difference.Also, we establish the stability in Lp-Shephard problem. To include the case wherep=n, we actually study the stability problem of the volume-normalized Lppolarprojection bodies of two convex bodies. To achieve this, we establish a new connectionbetween the Fourier analytic approach and the volume-normalized Lppolar projectionbody. Based on this connection, we give a complete answer to the stability of theLp-Shephard problem. These results not only contain the classical case of p=4k+1, butalso the case of p=4k-1. The case where p=n and p is an odd number also is included.