Geometrical Inequalities And Shephard Problem For Convex Body In The L_p-Space

Convex geometry is an important branch of modern geometry, convex bodiesand star bodies are the main research objects and Lp Brunn-Minkowski theory iskernel of convex body theory. This paper, on the one hand, uses basic knowledgeand approaches of Lp Brunn-Minkowski theory to consider some basic problems onthis theory. On the other hand, quasi-Shephard problem on classical projection bodyand Lp mixed projection body are researched by using Fourier analytic approach andMinkowski-Funk transform. Our main results can be stated as follows:(1) In1988, Lutwak frst explicitly introduced the concept of intersection bodies.In2005, Haberl and Ludwig extended classical intersection bodies, and posed thenotion of Lp intersection bodies. This paper introduces the concept of generalizedintersection bodies, and uses Lp Brunn-Minkowski theory, Ho¨lder integral inequalityand Minkowski integral inequality to consider monotonicity of generalized intersectionbodies, and establishes some Brunn-Minkowski inequalities for radial linear combina-tion and Lp harmonic Blaschke combination.(2) The classical projection body ΠK of convex body K in Rnwas introducedat the turn of the previous century by Minkowski. Combing the concept of classicalafne surface area, we develop a Fourier analytic approach to study the problemin the Brunn-Minkowski-Firey theory of convex bodies. We formulate and solve aquasi-Shephard problem on projections of convex bodies.(3) According to notion of Lp projection body ΠpK of convex body K in Rnintroduced by Lutwak, Yang and Zhang, we introduce concept of Lp mixed projec-tion body Πp,iK, i-type Lp afne surface area (i)p(K) of convex body K in RnandLp mixed curvature function fp,i(K,·), which characterizes the support function ofLp mixed projection body by Minkowski-Funk transform. Then a general solutionof quasi-Shephard problem for Lp mixed projection is given.