L p -Curvature Measures and L p , q -Mixed Volumes

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