DUAL Lp AFFINE ISOPERIMETRIC INEQUALITIES

Corresponding to each convex (or more general) subset of n-dimensional Euclidean space, Rn, there is a unique ellipsoid with the following property. The moment of inertia of the ellipsoid and the moment of inertia of the convex set are the same about every 1-dimensional subspace of Rn. This ellipsoid is called the Legendre ellipsoid of the convex set. The Legendre ellipsoid is a well-known concept from classical mechanics. For a star-shaped (about the origin) set K ⊂Rn, it is easy to see that its Legendre ellipsoid, usually denoted by Γ2K , is an object of the dual Brunn-Minkowski theory. In [6], the dual analog of the classical Legendre ellipsoid in the Brunn-Minkowski theory is introduced. For a convex body (i.e., a compact, convex subset with nonempty interior) K in Rn, its dual analog of Γ2K is dented by Γ−2K . More in general, in [8], the Lp analog of centroid bodies, ΓpK for a convex body K also being investigated, and, in [7], the dual of ΓpK , Γ−pK are defined. The main aim of this article is to establish some affine inequalities for Γ−pK , which are dual analog of the main results in [5, 8]. The techniques developed by Lutwak, Yang, and Zhang play a critical role throughout our paper. Let Sn−1 denote the unit sphere in Rn. Let B denote the unit ball (the convex hull of Sn−1) in Rn, and write ωn for the n-dimensional volume of B. Note that ωn = π n/2 Γ(1+n/2) (1.1)