For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn–Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are “equivalent” in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.

The $L_p$ cosine transform on Grassmann manifolds naturally induces finite dimensional Banach norms whose unit balls are origin-symmetric convex bodies in $\rn$. Reverse isoperimetric type volume inequalities for these bodies are established, which extend results from the sphere to Grassmann manifolds.

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

In this paper, the Lp(p\geq1) mean zonoid of a convex body K is given, and we show that it is the Lp centroid body of radial (n+p)th mean body of K up to a dilation. We also establish some affine inequalities of these bodies by proving that the volume of the new bodies is decreasing under Steiner symmetrization.

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.