In this paper, it is proved that the weak convergence of the L p Guassian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for p β₯ 1. Moreover, this paper obtains the solution to the L p Guassian Minkowski problem is continuous with respect to p.2000 Mathematics Subject Classification. 52A40. Key words and phrases. convex body; continuity; L p Gaussian surface area measure; L p Gaussian Minkowski problem.
The (π, π)-th dual curvature measure was introduced by Lutwak, Yang and Zhang. In this paper, we study the πΏ π cosine transform of the (π, π)-th dual curvature measure which defines a new convex body. We prove that the new convex body unifies the πΏ π Petty projection body and the πΏ π centroid body. The corresponding affine isoperimetric inequality is also obtained. It is an extension of the known πΏ π Petty projection inequality.
K E Y W O R D SπΏ π centroid body, πΏ π cosine transform, πΏ π Petty projection body, (π, π)-th dual curvature measure M S C ( 2 0 2 0 ) 52A20, 52A40
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