General affine surface areas

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for $L_{\phi}$ affine surface areas are established.Comment: Next version; minor change

On the uniqueness of LpL_p-Minkowski problems: the constant pp-curvature case in R3\mathbb{R}^3

We study the $C^4$ smooth convex bodies $\mathbb{K}\subset\mathbb{R}^{n+1}$ satisfying $K(x)=u(x)^{1-p}$, where $x\in\mathbb{S}^n$, $K$ is the Gauss curvature of $\partial\mathbb{K}$, $u$ is the support function of $\mathbb{K}$, and $p$ is a constant. In the case of $n=2$, either when $p\in[-1,0]$ or when $p\in(0,1)$ in addition to a pinching condition, we show that $\mathbb{K}$ must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the $L_p$-Minkowski problem in $\mathbb{R}^3$. Moreover, we give an explicit pinching constant depending only on $p$ when $p\in(0,1)$.Comment: references update

New LpL_p Affine Isoperimetric Inequalities

We prove new $L_p$ affine isoperimetric inequalities for all $p \in [-\infty,1)$. We establish, for all $p\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_\frac{n^2}{p}$ affine surface area of the polar body $K^\circ$

A direct proof of the functional Santalo inequality

We give a simple proof of a functional version of the Blaschke-Santalo inequality due to Artstein, Klartag and Milman. The proof is by induction on the dimension and does not use the Blaschke-Santalo inequality.Comment: 4 pages, file might be slighlty different from the published versio

On power means of positive quadratic forms

AbstractSome power means of positive definite quadratic forms are closely related to the fundamental scalar functions of the matrix associated with the quadratic form. This relation can (among other things) be used to give new proofs of some of the classical matrix inequalities

Renormalization of potentials and generalized centers

We generalize the Riesz potential of a compact domain in $\mathbb{R}^{m}$ by introducing a renormalization of the $r^{\alpha-m}$-potential for $\alpha\le0$. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.Comment: Adv. Appl. Math. 48 (2012), 365--392 Figure 11 has been corrected after publication. Theorem 3.12 and the exposition of Lemma 2.15 are modified in version