Results 1  10
of
86
Discrete tomography: determination of finite sets by Xrays
 Trans. Amer. Math. Soc
, 1997
"... Abstract. We study the determination of finite subsets of the integer lattice Z n, n ≥ 2, by Xrays. In this context, an Xray of a set in a direction u gives the number of points in the set on each line parallel to u. For practical reasons, only Xrays in lattice directions, that is, directions par ..."
Abstract

Cited by 43 (3 self)
 Add to MetaCart
Abstract. We study the determination of finite subsets of the integer lattice Z n, n ≥ 2, by Xrays. In this context, an Xray of a set in a direction u gives the number of points in the set on each line parallel to u. For practical reasons, only Xrays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are four prescribed lattice directions such that convex subsets of Z n (i.e., finite subsets F with F = Z n ∩ conv F) are determined, among all such sets, by their Xrays in these directions. We also show that three Xrays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammer’s Xray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in Z 2 have the property that convex subsets of Z 2 are determined, among all such sets, by their Xrays in these directions. We also consider the use of orthogonal projections in the interactive technique of successive determination, in which the information from previous projections can be used in deciding the direction for the next projection. We obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.
Affine Isoperimetric Inequalities
 J. Differential Geom
, 2002
"... this article deals with inequalities for centroid and projection bodies. Centroid bodies were attributed by Blaschke (see e.g., the books of Schneider [S2] and Leichtwei [Le] for references) to Dupin. If K is an originsymmetric convex body in Euclidean nspace, R n , then the centroid body of K i ..."
Abstract

Cited by 37 (0 self)
 Add to MetaCart
this article deals with inequalities for centroid and projection bodies. Centroid bodies were attributed by Blaschke (see e.g., the books of Schneider [S2] and Leichtwei [Le] for references) to Dupin. If K is an originsymmetric convex body in Euclidean nspace, R n , then the centroid body of K is the body whose boundary consists of the locus of the centroids of the halves of K formed when K is cut by codimension 1 subspaces. Blaschke (see Schneider [S2] for references) conjectured that the ratio of the volume of a body to that of its centroid body attains its maximum precisely for ellipsoids. This conjecture was proven by Petty [P1] who also extended the denition of centroid bodies and gave centroid bodies their name. When written as an inequality, Blaschke's conjecture is known as the BusemannPetty centroid inequality. Busemann's name is attached to the inequality because Pet
ANALYTIC SOLUTION TO THE BUSEMANNPETTY PROBLEM ON SECTIONS OF CONVEX BODIES
, 1999
"... We derive a formula connecting the derivatives of parallel section functions of an originsymmetric star body in Rn with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n − 1)dimensional Xray) gives the ((n − 1)dimensional) volumes of all hyp ..."
Abstract

Cited by 34 (7 self)
 Add to MetaCart
We derive a formula connecting the derivatives of parallel section functions of an originsymmetric star body in Rn with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n − 1)dimensional Xray) gives the ((n − 1)dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in Rn and leads to a unified analytic solution to the BusemannPetty problem: Suppose that K and L are two originsymmetric convex bodies in Rn such that the ((n − 1)dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (ndimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the BusemannPetty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n − 2)nd derivative of the parallel section functions. The affirmative answer to the BusemannPetty problem for n ≤ 4 and negative answer for n ≥ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.
A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES
 VOL. 104, NO. 3 DUKE MATHEMATICAL JOURNAL
, 2000
"... Corresponding to each originsymmetric convex (or more general) subset of Euclidean nspace R n, 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 1dimensional subspace of R n. This el ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
Corresponding to each originsymmetric convex (or more general) subset of Euclidean nspace R n, 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 1dimensional subspace of R n. This ellipsoid is called the Legendre ellipsoid of the convex set. The Legendre ellipsoid and its polar (the Binet ellipsoid) are wellknown concepts from classical mechanics. See Milman and Pajor [MPa1], [MPa2], Lindenstrauss and Milman [LiM], and Leichtweiß [Le] for some historical references. It has slowly come to be recognized that alongside the BrunnMinkowski theory there is a dual theory. The nature of the duality between the BrunnMinkowski theory and the dual BrunnMinkowski theory is subtle and not yet understood. It is easily seen that the Legendre (and Binet) ellipsoid is an object of this dual BrunnMinkowski theory. This observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical Legendre ellipsoid in the BrunnMinkowski theory. It is the aim of this paper to demonstrate the existence of precisely this dual object. In retrospect, one may well wonder why the new ellipsoid presented in this note was not discovered long ago. The simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of L2curvature in hand. However, the BrunnMinkowski theory was only recently extended to incorporate the new notion of Lpcurvature (see [L2], [L3]). A positivedefinite n × n real symmetric matrix A generates an ellipsoid ɛ(A), in R n, defined by ɛ(A) = { x ∈ R n: x ·Ax ≤ 1}, where x ·Ax denotes the standard inner product of x and Ax in Rn. Associated with a starshaped (about the origin) set K ⊂ Rn is its Legendre ellipsoid Ɣ2K, which is generated by the matrix [mij (K)] −1, where mij (K) = n+2 ei ·x
A BrunnMinkowski inequality for the integer lattice
 Trans. Amer. Math. Soc
"... Abstract. A close discrete analog of the classical BrunnMinkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
Abstract. A close discrete analog of the classical BrunnMinkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A RogersShephard type inequality for the lattice point enumerator in the plane is also proved. 1.
General Lp affine isoperimetric inequalities
 J. Differential Geometry
"... Projection bodies were introduced by Minkowski at the turn of the previous century and have since become a central notion in convex geometry. They arise ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
Projection bodies were introduced by Minkowski at the turn of the previous century and have since become a central notion in convex geometry. They arise
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, probab ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
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.
Moment inequalities and central limit properties of isotropic convex bodies
 Math. Zeitschr
, 2002
"... convex bodies ..."