On The L_p-Minkowski Problem For Homogeneous Measures

Based on the volume-normalized L_p-Minkowski problem,we extend volume measure to homogeneous measure.And in this paper,we investigate the existence of solutions to the normalized Lp-Minkowski problem ofμ-measures for the case p 1.The densities of the measures considered in this paper are r-concave,1/r-homogeneous(r>0).The normalized L_p-Minkowski problem ofμmeasures asks for necessary and sufficient conditions for a given Borel measure on the unit sphere so that it is the normalized L_p-surfaceμ-area measure of a convex body.To begin with,the existence of the solution to the L_p-Minkowski problem ofμ-measures for the case p 1 is studied,and the resulting convex body is origin-symmetric;Then,in the process of continuing to prove the solution of the problem,two new methods are used to prove the existence of the solution of the normalized L_p-Minkowski problem.