Some Geometry Inequalities For The Convex Body In The Lp-Space

This article belongs to the Lp-Brunn-Minkowski theory, which is a high-speeddeveloping geometry branch during the past decades, and devoted to study the convexbodies of Lp-space by using the basic notions，basic theories and integral transforms ofLp-Brunn-Minkowski theory.In chapter1, the background of the Lp-Brunn-Minkowski theory and main results aregiven. In chapter2-3, extension on some aspects of the basic theory ofLp-Brunn-Minkowski theory are obtained, based on the notions of Lp-affine surface area,Lp-geominimal surface area and Lp-dual affine surface area, the notions of Lp-dual mixedaffine surface area and Lp-dual geominimal surface area are introuduced, and theirproperty, Blaschke-Santalo inequality and Brunn-Minkowski inequality are studied. Fromchapter4to chapter7, the Lp-intersection bodies, the new geometry bodies, Lp-mixedprojection bodies, the general Lp-projection bodies and the general Lp-mixed projectionbodies are reaserched. For the Lp-intersection bodies, some special Brunn-Minkowskiinequalities are obtained by combing with Lp-harmonic radial combination andLp-harmonic Blaschke combination. For the new geometry bodyΓ_-pK, the negative formof the Shephard problem and the Lp-affine surface area form of the Shephard problem aregiven. For the Lp-mixed projection bodies, some properties are given by using Lp-mixedquermassintegrals, futher, the negative form of the Shephard problem are studied. For thegeneral Lp-projection body, Lp-mixed volume and Lp-mixed affine surface area forms ofthe Shephard problem are given. For the general Lp-mixed projection body, the pettyprojection inequality and quermassintegrals form of Shephard problem are obtained.