Obervational Model for Microarcsecond Astrometry with the Space Interferometry Mission

The Space Interferometry Mission (SIM) is a space-based long-baseline optical interferometer for precision astrometry. One of the primary objectives of the SIM instrument is to accurately determine the directions to a grid of stars, together with their proper motions and parallaxes, improving a priori knowledge by nearly three orders of magnitude. The basic astrometric observable of the instrument is the pathlength delay, a measurement made by a combination of internal metrology measurements that determine the distance the starlight travels through the two arms of the interferometer and a measurement of the white light stellar fringe to find the point of equal pathlength. Because this operation requires a non--negligible integration time to accurately measure the stellar fringe position, the interferometer baseline vector is not stationary over this time period, as its absolute length and orientation are time--varying. This conflicts with the consistency condition necessary for extracting the astrometric parameters which requires a stationary baseline vector. This paper addresses how the time-varying baseline is ``regularized'' so that it may act as a single baseline vector for multiple stars, and thereby establishing the fundamental operation of the instrument.Comment: 24 pages, 6 figure

Dual Mixed Volumes and the Slicing Problem

We develop a technique using dual mixed-volumes to study the isotropic constants of some classes of spaces. In particular, we recover, strengthen and generalize results of Ball and Junge concerning the isotropic constants of subspaces and quotients of L_p and related spaces. An extension of these results to negative values of p is also obtained, using generalized intersection-bodies. In particular, we show that the isotropic constant of a convex body which is contained in an intersection-body is bounded (up to a constant) by the ratio between the latter's mean-radius and the former's volume-radius. We also show how type or cotype 2 may be used to easily prove inequalities on any isotropic measure.Comment: 38 pages, to appear in Advances in Mathematics. Corrected Remark 4.

Generalized Intersection Bodies

We study the structures of two types of generalizations of intersection-bodies and the problem of whether they are in fact equivalent. Intersection-bodies were introduced by Lutwak and played a key role in the solution of the Busemann-Petty problem. A natural geometric generalization of this problem considered by Zhang, led him to introduce one type of generalized intersection-bodies. A second type was introduced by Koldobsky, who studied a different analytic generalization of this problem. Koldobsky also studied the connection between these two types of bodies, and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized Busemann-Petty problem. We show that these classes share many identical structure properties, proving the same results using Integral Geometry techniques for Zhang's class and Fourier transform techniques for Koldobsky's class. Using a Functional Analytic approach, we give several surprising equivalent formulations for the equivalence problem, which reveal a deep connection to several fundamental problems in the Integral Geometry of the Grassmann Manifold.Comment: 45 pages, to appear in Journal of Functional Analysis Revised version after referee's comment

Generalized Intersection Bodies are not Equivalent

In 2000, A. Koldobsky asked whether two types of generalizations of the notion of an intersection-body, are in fact equivalent. The structures of these two types of generalized intersection-bodies have been studied by the author in [http://www.arxiv.org/math.MG/0512058], providing substantial positive evidence for a positive answer to this question. The purpose of this note is to construct a counter-example, which provides a surprising negative answer to this question in a strong sense. This implies the existence of non-trivial non-negative functions in the range of the spherical Radon transform, and the existence of non-trivial spaces which embed in L_p for certain negative values of p.Comment: 18 pages, added a section with equivalent formulations using Fourier Transforms and Embeddings into L_p for p<