Abstract
For star bodies, the dual affine quermassintegrals were introduced and studied in several papers. The aim of this paper is to study them further. In this paper, some inequalities for dual affine quermassintegrals are established, such as the Minkowski inequality, the dual Brunn-Minkowski inequality, and the Blaschke-Santaló inequality.
Highlights
The setting for this paper is n-dimensional Euclidean space Rn
Affine quermassintegrals are important geometric invariants related to the projection of convex body
In [6], Lutwak introduced the dual affine quermassintegrals of a star body L containing the origin in its interior, Φi(L), by letting Φ0(L) = V (L), Φn(L) = kn, and for 0 < i < n, Φi(L) = kn
Summary
The setting for this paper is n-dimensional Euclidean space Rn. Let n denote the set of convex bodies (compact, convex subsets with nonempty interiors) and on denote the subset of n that consists of convex bodies with the origin in their interiors. Affine quermassintegrals are important geometric invariants related to the projection of convex body These quermassintegrals were introduced by Lutwak [7], and can be defined by letting Φ0(K) = V (K), Φn(K) = kn, and for 0 < i < n, Φi(K ) = kn voln−i K | ξ. Where the Grassmann manifold G(n,i) is endowed with the normalized Haar measure, and kn is the volume of the unit ball Bn in Rn. in [6], Lutwak introduced the dual affine quermassintegrals of a star body L containing the origin in its interior, Φi(L), by letting Φ0(L) = V (L), Φn(L) = kn, and for 0 < i < n, Φi(L) = kn voln−i(L ∩ ξ) kn−i n dξ.
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