Lp -Curvature Measures and Lp,q -Mixed Volumes

Abstract

Motivated by Lutwak et al.'s Lp-dual curvature measures, we introduce the concept of Lp-curvature measures. This new Lp-curvature measure is an extension of the classical surface area measure, Lp-surface area measure, and curvature measure. In this paper, we first prove some properties of the Lp-curvature measure. Next, using the Lp-curvature measure, we define the Lp,q-mixed volume which includes Lp-mixed volume as the special cases. Further, the Minkowski-type inequality related Lp,q-mixed volume and the uniqueness of the solution for the Lp,q- Minkowski problem are obtained. Finally, we propose several problems that need to be studied further.