This paper deals with Lp geominimal surface area and its extension to Lp mixed geominimal surface area. We give an integral formula of Lp geominimal surface area by the p-Petty body and introduce the concept of Lp mixed geominimal surface area which is a natural extension of Lp geominimal surface area. Some inequalities, such as, analogues of Alexandrov–Fenchel inequalities, Blaschke–Santaló inequalities, and affine isoperimetric inequalities for Lp mixed geominimal surface areas are obtained.

Corresponding to the Legendre ellipsoid and the LYZ ellipsoid, two new sine ellipsoids are introduced in this paper. These four ellipsoids are closely related in the Pythagorean relation and duality. Several volume inequalities and the valuation properties are obtained for two new ellipsoids.

In this article, a classification of continuous, linearly intertwining, symmetric $L_p$-Blaschke ($p>1$) valuations is established as an extension of Haberl's work on Blaschke valuations. More precisely, we show that for dimensions $n \geq 3$, the only continuous, linearly intertwining, normalized symmetric $L_p$-Blaschke valuation is the normalized $L_p$-curvature image operator, while for dimension $n = 2 $, a rotated normalized $L_p$-curvature image operator is an only additional one. One of the advantages of our approach is that we deal with normalized symmetric $L_p$-Blaschke valuations, which makes it possible to handle the case $p=n$. The cases where $p \neq =n$ are also discussed by studying the relations between symmetric $L_p$-Blaschke valuations and normalized ones.

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.

Sufficient conditions are given for the existence of solutions to the discrete LpMinkowski problem for p-capacity when 0 <p <1and 1 <p <2.