828 research outputs found
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 affine surface areas are established.Comment: Next version; minor change
On the uniqueness of -Minkowski problems: the constant -curvature case in
We study the smooth convex bodies
satisfying , where , is the Gauss
curvature of , is the support function of ,
and is a constant. In the case of , either when or when
in addition to a pinching condition, we show that must
be the unit ball. This partially answers a conjecture of Lutwak, Yang, and
Zhang about the uniqueness of the -Minkowski problem in .
Moreover, we give an explicit pinching constant depending only on when
.Comment: references update
New Affine Isoperimetric Inequalities
We prove new affine isoperimetric inequalities for all . We establish, for all , a duality formula which shows
that affine surface area of a convex body equals
affine surface area of the polar body
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 by
introducing a renormalization of the -potential for .
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
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