The quermassintegral inequalities for k-convex starshaped domains

We give a proof of the isoperimetric inequality for quermassintegrals of non-convex starshaped domains, using a result of Gerhardt [C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geometry 32 (1990) 299–314] and Urbas [J. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990) 355–372] on an expanding geometric curvature flow.