In this paper, we develop a Fourier analytic approach to study the problem in the Brunn-Minkowski-Firey theory of convex bodies. We formulate and solve a quasi-Shephard's problem on projections of convex bodies.
In this article, we study the convex bodies associated with Lp-projections in the Brunn-Minkowski-Firey theory, and apply the Fourier analytic methods to prove the linear stability in the Shephard problem for Lp-projections of convex bodies.
Zhu,Lü and Leng extended the concept of L_p-polar curvature image. We continuously study the L_p-polar curvature image and mainly expound the relations between the volumes of star bodies and their L_p-polar curvature images in this article. We first establish the L_p-affine isoperimetric inequality associated with L_p-polar curvature image. Secondly,we give a monotonic property for L_p-polar curvature image. Finally, we obtain an interesting equation related to L_p-projection body of L_p-polar curvature image and L_p-centroid body.
According to the notion of Orlicz mixed volume, in this paper, we extend L_p-dual affine surface area to the Orlicz version. Further, we obtain the affine isoperimetric inequality and the Blachke-Santaló inequality for the dual Orlicz affine surface area. Besides, we also get the monotonicity inequality for Orlicz dual affine surface area.
