Isotropic surface area measures

Abstract

The purpose of this note is to bring into attention an apparently forgotten result of C. M. Petty: a convex body has minimal surface area among its affine transformations of the same volume if, and only if, its area measure is isotropic. We obtain sharp affine inequalities which demonstrate the fact that this “surface isotropic” position is a natural framework for the study of hyperplane projections of convex bodies.

Volume inequalities for complex isotropic measures

“…. , m. These inequalities have had a profound impact on convex geometric analysis (see, e.g., [6][7][8][9][10][11][12]17,18]). Using the same notations, we write P span{u,u † } and P [span{u,u † }] ⊥ , u ∈ S 2n−1 , for the orthogonal projections onto the 2-dimensional subspace span{u, u † } and the (2n − 2)-dimensional subspace [span{u, u † }] ⊥ , respectively.…”

Isotropic dual p-surface area measure

“…From (1) and the definition of L p -harmonic radial combination, it follows immediately that for an L pharmonic radial combination of star bodies K and L,…”

“…1 Isotropic dual p-surface area measure Giannopoulos and Papadimitrakis [1] pointed out that a convex body has minimal surface area among its volume preserving affine transformations if and only if its area measure is isotropic. Motivated by their work, we studied the dual p-surface area which was introduced by Lutwak, et al [2] and got a similar result.…”

VOLUME INEQUALITIES FOR THE Lp-SINE TRANSFORM OF ISOTROPIC MEASURES

“…, b n denote orthonormal basis vectors of R n . Isotropic measures have been the focus of recent studies, in particular, in relation with a variety of extremal problems for convex bodies (see, e.g., [8,9,16,20,29,43,45] and the references therein).…”