Abstract
We establish some inequalities for the dual -centroid bodies which are the dual forms of the results by Lutwak, Yang, and Zhang. Further, we establish a Brunn-Minkowski-type inequality for the polar of dual -centroid bodies.
Highlights
Corresponding to each convex subset of n-dimensional Euclidean space, Rn, there is a unique ellipsoid with the following property
The Legendre ellipsoid is a well-known concept from classical mechanics
For a star-shaped 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
Summary
Corresponding to each convex (or more general) subset of n-dimensional Euclidean space, Rn, there is a unique ellipsoid with the following property. We establish the weak dual analog of Theorem 1.3 for Γ−pK and get the following inequality. If K is a convex body in Rn, its support function hK (·) : Sn−1 → R is defined by hK (u) = max{u · x : x ∈ K}.
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