Extremum Problems And Stability In Convex Geometric

In this article, we discuss the hot topics on stability of convex bodies in Lp-Brunn-Minkowski theory, and application the Fourier transform and isometric imbedding made a further study of geometry for the measure properties, such as Lp-projection body, Lp-affine surface area on the basis of predecessors, and got some inequalities related to the stability results. Then, combination with the concept of Lp-polar curvature image and discussed the mononicity of Lp-polar curvature image, ob-tained the affine isoperimetric inequality of Lp-polar curvature image. In addi-tion, we established (i,j)-typeLp-mixed affine isoperimetric inequality and (i,j)-typeLp-mixed affine isoperimetric inequality. Finally, we introduced a concept of Lfaq]-mixed affine surface area and made a further promotion for Lp-affine isoperi-metric inequality by the Holder inequality.