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54
On some affine isoperimetric inequalities,
 J. Differential Geom.
, 1986
"... Abstract The Lp analogues of the Petty projection inequality and the BusemannPetty centroid inequality are established. An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under nondegenerate l ..."
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Cited by 113 (5 self)
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Abstract The Lp analogues of the Petty projection inequality and the BusemannPetty centroid inequality are established. An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under nondegenerate linear transformations. These isoperimetric inequalities are more powerful than their betterknown Euclidean relatives. This article deals with affine isoperimetric inequalities for centroid and projection bodies. Centroid bodies were attributed by Blaschke to Dupin (see e.g., the books of Schneider
Counting the faces of randomlyprojected hypercubes and orthants, with applications
, 2008
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Multivariate Gini Indices
 Journal of Multivariate Analysis
, 1995
"... The Gini index and the Gini mean difference of a univariate distribution are extended to measure the disparity of a general dvariate distribution. We propose and investigate two approaches, one based on the distance of the distribution from itself, the other on the volume of a convex set in (d + 1) ..."
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Cited by 22 (6 self)
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The Gini index and the Gini mean difference of a univariate distribution are extended to measure the disparity of a general dvariate distribution. We propose and investigate two approaches, one based on the distance of the distribution from itself, the other on the volume of a convex set in (d + 1)space, named the lift zonoid of the distribution. When d = 1, this volume equals the area between the usual Lorenz curve and the line of zero disparity, up to a scale factor. We get two definitions of the multivariate Gini index, which are different (when d#1) but connected through the notion of the lift zonoid. Both notions inherit properties of the univariate Gini index, in particular, they are vector scale invariant, continuous, bounded by 0 and 1, and the bounds are sharp. They vanish if and only if the distribution is concentrated at one point. The indices have aceteris paribus property and are consistent with multivariate extensions of the Lorenz order. Illustrations with data conclude...
Rotation equivariant Minkowski valuations
 INT. MATH. RES. NOT. (2006), ARTICLE ID 72894
"... The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π ..."
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Cited by 18 (7 self)
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The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only nontrivial operator with these properties.
Inverse formula for the BlaschkeLevy representation
 Houston J. Math
, 1997
"... Abstract. We say that an even continuous function H on the unit sphere Ω in Rn admits the BlaschkeLevy representation with q> 0 if there exists an even function b ∈ L1(Ω) so that Hq (x) = ∫ Ω (x, ξ)qb(ξ) dξ for every x ∈ Ω. This representation has numerous applications in convex geometry, pro ..."
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Cited by 16 (1 self)
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Abstract. We say that an even continuous function H on the unit sphere Ω in Rn admits the BlaschkeLevy representation with q> 0 if there exists an even function b ∈ L1(Ω) so that Hq (x) = ∫ Ω (x, ξ)qb(ξ) dξ for every x ∈ Ω. This representation has numerous applications in convex geometry, probability and Banach space theory. In this paper, we present a simple formula (in terms of the derivatives of H) for calculating b out of H. We use this formula to give a sufficient condition for isometric embedding of a space into Lp which contributes to the 1937 P.Levy’s problem and to the study of zonoids. Another application gives a Fourier transform formula for the volume of (n − 1)dimensional central sections of star bodies in Rn. We apply this formula to find the minimal and maximal volume of central sections of the unit balls of the spaces ℓn p with 0 < p < 2. 1.
Valuations and BusemannPetty type problems
 Adv. Math
"... Projection and intersection bodies define continuous and GL(n) contravariant valuations. They played a critical role in the solution of the Shephard problem for projections of convex bodies and its dual version for sections, the Busemann– Petty problem. We consider the question whether ΦK ⊆ ΦL impli ..."
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Cited by 12 (2 self)
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Projection and intersection bodies define continuous and GL(n) contravariant valuations. They played a critical role in the solution of the Shephard problem for projections of convex bodies and its dual version for sections, the Busemann– Petty problem. We consider the question whether ΦK ⊆ ΦL implies V (K) ≤ V (L), where Φ is a homogeneous, continuous operator on convex or star bodies which is an SO(n) equivariant valuation. Important previous results for projection and intersection bodies are extended to a large class of valuations.