The Continuity Of The Solution To The Dual Minkowski Problem

Convex geometry is the modern geometric subject that studies geometric structures and invariants of geometric bodies(which mainly include convex bod-ies and star bodies)by using both geometric and analytic methods.The Brunn-Minkowski theory is the very core of convex geometric analysis,and the Minkows-ki problem is one of the most fundamental problems of the Brunn-Minkowski theory.The Minkowski problem has always been one of the hot issues in inte-gral geometry and convex geometry,differential geometry,geometric analysis and PDE.It mainly involves the existence,uniqueness,continuity and regularity of solutions.In this thesis,we study mainly continuity of the solution to the dual Minkowski problem.Firstly,we consider continuity of the solution to the log-Minkowski problem which is just the L_pMinkowski problem when p=0 and the dual Minkowski problem when q=n.When p>1 with p?n,G.Zhu solved continuity of the solution to the L_pMinkowski problem.Motivated by Zhu's works,we consider continuity of the solution to the log-Minkowski problem.In the plane R~2,by the log-Minkowski inequality and uniqueness of the solution to the log-Minkowski problem,we obtain continuity of the solution to the even log-Minkowski problem.Without the even data,we also obtain it is not correct in R~n.Secondly,we study continuity of the solution to the dual Minkowski problem for negative indices.The dual Minkowski problem is not only a generalization of the log-Minkowski problem but also the“dual”case of the L_pMinkowski problem.According to the continuity of solution of L_pMinkowski problem,we naturally consider continuity of the solution to the dual Minkowski problem.By the opti-mization problem associated with the dual Minkowski problem and uniqueness of the solution to the dual Minkowski problem,we obtain the dual log-Minkowski inequality for the dual curvature measure.Then,from uniqueness of the solution to the dual Minkowski problem and the dual log-Minkowski inequality,we obtain continuity of the solution to the dual Minkowski problem for negative indices.Thirdly,we discuss the dual Minkowski problem for the dual area measure.By the cyclic inequality for the dual mixed volume and the dual log-Minkowski inequality for the dual cone-volume measure,we obtain uniqueness of the solution to the dual Minkowski problem for the dual area measure.Based on this result,we prove continuity of the solution to the dual Minkowski problem for q>n-1.What's more,we obtain the results for the stabilities of the dual log-Minkowski inequality and the dual log-Brunn-Minkowski inequality.