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14
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
AFFINE INEQUALITIES AND RADIAL MEAN BODIES
, 1998
"... We introduce for p> 1 the radial pth mean body RpK of a convex body K in En. The distance from the origin to the boundary of RpK in a given direction is the pth mean of the distances from points inside K to the boundary of K in the same direction. The bodies RpK form a spectrum linking the diff ..."
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Cited by 23 (2 self)
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We introduce for p> 1 the radial pth mean body RpK of a convex body K in En. The distance from the origin to the boundary of RpK in a given direction is the pth mean of the distances from points inside K to the boundary of K in the same direction. The bodies RpK form a spectrum linking the difference body of K and the polar projection body of K, which correspond to p = 1 and p = 1, respectively. We prove that RpK is convex when p> 0. We also establish a strong and sharp affine inequality relating the volume of RpK to that of RqK when 1 < p < q. When p = n and q!1, this becomes the RogersShephard inequality, and when p! 1 and q = n, it becomes a reverse form of the Petty projection inequality proved previously by the second author.
and Minkowskiendomorphisms of convex bodies
 Trans. Amer. Math. Soc
"... Abstract. We consider maps of the family of convex bodies in Euclidean ddimensional space into itself that are compatible with certain structures on this family: A Minkowskiendomorphism is a continuous, Minkowskiadditive map that commutes with rotations. For d ≥ 3, a representation theorem for su ..."
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Cited by 18 (0 self)
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Abstract. We consider maps of the family of convex bodies in Euclidean ddimensional space into itself that are compatible with certain structures on this family: A Minkowskiendomorphism is a continuous, Minkowskiadditive map that commutes with rotations. For d ≥ 3, a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by applying the generalized spherical Radon transform to support functions. We give a complete characterization of weakly monotonic Minkowskiendomorphisms. A corresponding theory is developed for Blaschkeendomorphisms, where additivity is now understood with respect to Blaschkeaddition. Using a special mixed volume, an adjoining operator can be introduced. This operator allows one to identify the class of Blaschkeendomorphisms with the class of weakly monotonic, nondegenerate and translationcovariant Minkowskiendomorphisms. The following application is also shown: If a (weakly monotonic and) nontrivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball. 1. Introduction and
Convolutions and multiplier transformations of convex bodies
 Trans. Amer. Math. Soc
"... Abstract. Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on BlaschkeMinkowski homomorphisms. We show that such maps are represented by a spherical con ..."
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Cited by 11 (4 self)
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Abstract. Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on BlaschkeMinkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even BlaschkeMinkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even BlaschkeMinkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator. Key words. Convex bodies, Minkowski addition, Blaschke addition, rotation intertwining map, spherical convolution, spherical harmonic, multiplier transformation,
A New Affine Invariant For Polytopes And Schneider's Projection Problem
, 2001
"... New ane invariant functionals for convex polytopes are introduced. Some sharp ane isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new ane isoperimetric inequalities are extensions of B ..."
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Cited by 9 (0 self)
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New ane invariant functionals for convex polytopes are introduced. Some sharp ane isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new ane isoperimetric inequalities are extensions of Ball's reverse isoperimetric inequalities. If K is a convex body (i.e., a compact, convex subset with nonempty interior) in Euclidean nspace, R n , then on the unit sphere, S n 1 , its support function, h(K; ) : S n 1 ! R, is dened for u 2 S n 1 by h(K; u) = maxfu y : y 2 Kg, where u y denotes the standard inner product of u and y. The projection body, K, of K can be dened as the convex body whose support function, for u 2 S n 1 , is given by h(K;u) = vol n 1 (Kju ? ); where vol n 1 denotes (n 1)dimensional volume and Kju ? denotes the image of the orthogonal projection of K onto the codimension 1 subspace orthogonal to u. An important unsolved problem regarding projection bodies is Schneider's projection problem: What is the least upper bound, as K ranges over the class of originsymmetric convex bodies in R n , of the aneinvariant ratio () [V (K)=V (K) n 1 ] 1=n ; 1991 Mathematics Subject Classication. 52A40. Key words and phrases. ane isoperimetric inequalities, reverse isoperimetric inequalities, projection bodies, asymptotic inequalities. Research supported, in part, by NSF Grant DMS{9803261 Typeset by A M ST E X 1 2 where V is used to abbreviate vol n . See [S1], [S2], [SW] and [Le]. Schneider [S1] conjectured that this ratio is maximized by parallelotopes. In [S1], Schneider also presented applications of such results in stochastic geometry. However, a counterexample was produced in [Br] to show that this is not the case. ...
Hessian measures of semiconvex functions and applications to support measures of convex bodies
"... ..."
Volume Inequalities and Additive Maps of Convex Bodies
"... Analogs of the classical inequalities from the Brunn Minkowski Theory for rotation intertwining additive maps of convex bodies are developed. We also prove analogs of inequalities from the dual Brunn Minkowski Theory for intertwining additive maps of star bodies. These inequalities provide generali ..."
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Cited by 6 (2 self)
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Analogs of the classical inequalities from the Brunn Minkowski Theory for rotation intertwining additive maps of convex bodies are developed. We also prove analogs of inequalities from the dual Brunn Minkowski Theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary we obtain a new Brunn Minkowski inequality for the volume of polar projection bodies.
On zonoids whose polars are zonoids
 Israel J. Math
, 1997
"... Abstract. Zonoids whose polars are zonoids, cannot have proper faces other than vertices or facets. However, there exist non–smooth zonoids whose polars are zonoids. Examples in R 3 and R 4 are given. ..."
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Cited by 4 (1 self)
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Abstract. Zonoids whose polars are zonoids, cannot have proper faces other than vertices or facets. However, there exist non–smooth zonoids whose polars are zonoids. Examples in R 3 and R 4 are given.