The L_p-Brunn-Minkowski Inequality For Volume Differences

The classic Brunn-Minkowski inequality says that V(tK)1/n is concave function with respect to t>0.This was extended to volume difference of convex bodies by prof.Leng.However,the volume difference for the important Lp Brunn-Minkowski inequality due to Lutwak remains open.This thesis is devoted to solve this question.In particular,we shall establish the following inequalities:suppose that K,L and D are compact domains of Rn,D' is a dilate copy of D.The equality holds if and only if K and L are dilate,and(V(K),V(D))=?(V(L),V(D')),where ? is a constant.For the well known dual Brunn-Minkowski inequality,there is also an Lp-version.We also present a volume difference version of this inequality:suppose that K,L and M are star bodies in Rn,M' is a dilate copy of M.The equality holds if and only if K and L are dilate,and(V(K),V(M))=?(V(L),V(M')),where ? is a constant.