q-dual mixed volumes and L p -intersection bodies

In this paper, we introduce the new notions of {L_{p}} -intersection and mixed intersection bodies. Inequalities for the q-dual volume sum of {L_{p}} -mixed intersection bodies are established.

Orlicz dual affine quermassintegrals

In the paper, our main aim is to generalize the dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating the first-order variation of the dual affine quermassintegrals, and call it the Orlicz dual affine quermassintegral. The fundamental notions and conclusions of the dual affine quermassintegrals and the Minkoswki and Brunn–Minkowski inequalities for them are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions. The new Orlicz–Minkowski and Orlicz–Brunn–Minkowski inequalities in a special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn–Minkowski inequality, which also imply the {L_{p}} -dual Minkowski inequality and Brunn–Minkowski inequality for the dual affine quermassintegrals.

Lp-DUAL MIXED GEOMINIMAL SURFACE AREA

Lutwak (Adv. Math., vol. 118(2), 1996, pp. 244–294) defined the notion of Lp-geominimal surface area based on Lp-mixed volumes. Recently, Wang and Qi (J. Inequal. Appl., vol. 2011, 2011, pp. 1–10) introduced the concept of Lp-dual geominimal surface area based on Lp-dual mixed volumes. In this paper, based on Lp-dual mixed quermassintegrals, we define the concept of Lp-dual mixed geominimal surface area and establish several inequalities for this new notion.