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120
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 ..."
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Cited by 42 (3 self)
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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.
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 ..."
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Cited by 15 (3 self)
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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 ..."
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Cited by 13 (3 self)
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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.
Reconstruction of Convex Bodies from Brightness Functions
, 2003
"... Algorithms are given for reconstructing an approximation to an unknown convex body from finitely many values of its brightness function, the function giving the volumes of its projections onto hyperplanes. One of these algorithms constructs a convex polytope with less than a prescribed number of ..."
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Cited by 11 (5 self)
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Algorithms are given for reconstructing an approximation to an unknown convex body from finitely many values of its brightness function, the function giving the volumes of its projections onto hyperplanes. One of these algorithms constructs a convex polytope with less than a prescribed number of facets, while the others do not restrict the number of facets. Convergence of the polytopes to the body is proved under certain essential assumptions including origin symmetry of the body. Also described is an oraclepolynomialtime algorithm for reconstructing an approximation to an originsymmetric rational convex polytope of fixed and full dimension that is only accessible via its brightness function. Some of the algorithms have been implemented, and sample reconstructions are provided.
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 11 (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.
Moment inequalities and central limit properties of isotropic convex bodies
 Math. Zeitschr
, 2002
"... convex bodies ..."