Extremal Problems In Convex Bodies Geometry

This thesis works for theoretical study on isoperimetric problem and related inequalities by using theory of geometry analysis, way of integral transformation and theory of analysis inequalities. First, we study from the following several sides: width integral and affine surface of convex bodies, equivalence of some classical inequalities in convex bodies geometry, extremal properties of projection bodies and intersection bodies, extremal problem of dual Quermassintegral for star bodies, extremal properties of polars for mixed projection bodies, extremal properties of dual Quermassintegral differences for projection bodies and intersection bodies, dual properties of mixed projection bodies and mixed intersection bodies. Secand, we establish polars forms and dual forms of the classical Minkowski inequality, Brunn-Minkowsk inequality and Aleksandrov-Fenchel inequality and prove some interrelated results, using analysis inequalities such as Holder integral inequality, Bellman inequality, Minkowski integral inqualitiy, Pachpatte inequality, Hilbert integral inequality and et al. As a very important area of geometry analysis, the geometry theory of convex bodies is widely applied in mathematical economics, stochastic geometry, stereology and the theory of information and et al.Our main resuls can be stated as follows:(1) we establish Aleksandrov-Fenchel inequality for polars of mixed projection bodies. Solving a important geometry problem of convex bodies which was followed with interest by Lutwak in 1988. Then, we improve some important results of polars for mixed projection bodies which was given by Lutwak.(2) In 2004, G. S. Leng first introduces the Quermassintegral difference function of convex bodies: If K, D ∈ Kⁿ and D (?) K, then Quermassintegral difference function of convex bodies K and D was difined: Dw_i(K,D) = W_i(K) -W_i(D), (0 ≤ i ≤ n - 1), and established Minkowski inequality and Brunn-Minkowski inequality for the Quermassintegral difference of convex bodies.Similarly, we introduce a new interrelated conpect—the concept of dual Quermassintegral sum for star bodies: If K,D ∈φⁿ, then dual Quermassintegral sum function of star bodies K and D is difined as follows: S_Wi(K,D) = W_i(K) + W_i(-D), (0 ≤ i≤ n - 1). If i = 0, then S_V(K, D) = V(K) + V(D), we call dual volume sum function of star bodies K and D.Further, we estabalish Minkowski inequality of dual Quermassintegral sum for mixed intersection bodies. It is just a general form of classical Minkowski inequality. On the other hand, we prove a strengthening form of classical Brunn-Minkowski inequality.(3) We introduce the concept of dual Quermassintegral difference of convex bodies and star bodies: If K ∈ Kⁿ, D ∈ φⁿ and D (?) K, then we difine the conpect of dual Quermassintegral difference of convex body K and star body D: D_Wi(K, D) = W_i(K) - W_i(D), 0 ≤ i ≤ n - 1.