On Some Dual Lq Brunn-Minkowski Type Inequalities

In this thesis,the dual Lq Brunn-Minkowski type inequalities in integral geom-etry and convex geometry is discussed.We obtain an inequality related to the Gauss measure(Theorem 3.2.1)by using the q-order weighted average and the Minkowski integral inequality.Then by the dual Lq transference principle,we investigate the dual Lq Brunn-Minkowski type inequality for dual mixed volume(Theorem 4.1.2),the dual Lq Brunn-Minkowski type inequality for dual affine quermassintegral(The-orem 4.1.4),the dual Lq Brunn-Minkowski type inequality for dual harmonic quer-massintegral(Theorem 4.1.6),and the dual Lq Brunn-Minkowski type inequality for intersection bodies(Theorem 4.1.8).Simplified proofs of these inequalities are given.This thesis is composed of two parts.The first part contain generalizations of the existing results for the dual Lq Brunn-Minkowski inequalities in integral and convex geometry.We introduce dual Lq Brunn-Minkowski type inequalities.We introduce the dual Lq transference principle,a new method to deal with the dual Lq Brunn-Minkowski inequalities.And a simplified proof of the dual Lq Brunn-Minkowski type inequalities of dual mixed volume is given by the dual Lq transfer-ence principle.In the second part,the Gauss measure is introduced.We generalize the Brunn-Minkowski inequality for the Gauss measure to dual Lq radial addition and obtain a new inequality for the Gauss measure.