The Fourier transform and Firey projections of convex bodies (original) (raw)
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Fourier transform and Firey projections of convex bodies
Indiana University Mathematics Journal, 2004
In this paper we develop a Fourier analytic approach to problems in the Brunn-Minkowski-Firey theory of convex bodies. We study the notion of Firey projections and prove a version of Aleksandrov's projection theorem. We also formulate and solve an analog of the Shephard problem for Firey projections.
Fourier Analytic Methods in the Study of Projections and Sections of Convex Bodies
Fourier Analysis and Convexity, 2004
It has been noticed long ago that many results on sections and projections are dual to each other, though methods used in the proofs are quite different and don't use the duality of underlying structures directly. In the paper [KRZ], the authors attempted to start a unified approach connecting sections and projections, which may eventually explain these mysterious connections. The idea is to use the recently developed Fourier analytic approach to sections of convex bodies (a short description of this approach can be found in [K7]) as a prototype of a new approach to projections. The first results seem to be quite promising. The crucial role in the Fourier approach to sections belongs to certain formulas connecting the volume of sections with the Fourier transform of powers of the Minkowski functional. An analog of these formula for the case of projections was found in [KRZ] and connects the volume of projections to the Fourier transform of the curvature function. This formula was applied in [KRZ] to give a new proof of the result of Barthe and Naor on the extremal projections of l p -balls with p > 2, which is similar to the proof of the result on the extremal sections of l p -balls with 0 < p < 2 in [K5]. Another application is to the Shephard problem, asking whether bodies with smaller hyperplane projections necessarily have smaller volume. The problem was solved independently by Petty and Schneider, and the answer is affirmative in the dimension two and negative in the dimensions three and higher. The paper [KRZ] gives a new Fourier analytic solution to this problem that essentially follows the Fourier analytic solution to the Busemann-Petty problem (the projection counterpart of Shephard's problem) from [K3]. The transition in the Busemann-Petty problem occurs between the dimensions four and five. In Section 4, we show that the transition in both problems has the same explanation based on similar Fourier analytic characterizations of intersection and projection bodies.
Projections of convex bodies and the fourier transform
Israel Journal of Mathematics, 2004
The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections of l p -balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections of l p -balls, and give a Fourier analytic solution to Shephard's problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function.
Firey linear combinations of convex bodies
Journal of Shanghai University (English Edition), 2009
For convex bodies, the Firey linear combinations were introduced and studied in several papers. In this paper the mean width of the Firey linear combinations of convex bodies is studied, and the lower bound of the mean width of the Firey linear combinations of convex body and its polar body is given.
An Analytic Solution to the Busemann-Petty Problem on Sections of Convex Bodies
The Annals of Mathematics, 1999
We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n − 1)dimensional X-ray) 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 R n and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in R n 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 (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty 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 Busemann-Petty problem for n ≤ 4 and the 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.
Some extensions of the class of convex bodies
2008
We introduce and study a new class of eps\epseps-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how eps\epseps-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, and find applications to geometric tomography. We introduce the notion of a circular projection
Springer INdAM Series, 2013
We investigate some basic properties of the heart ♥(K) of a convex set K. It is a subset of K, whose definition is based on mirror reflections of euclidean space, and is a non-local object. The main motivation of our interest for ♥(K) is that this gives an estimate of the location of the hot spot in a convex heat conductor with boundary temperature grounded at zero. Here, we investigate on the relation between ♥(K) and the mirror symmetries of K; we show that ♥(K) contains many (geometrically and phisically) relevant points of K; we prove a simple geometrical lower estimate for the diameter of ♥(K); we also prove an upper estimate for the area of ♥(K), when K is a triangle.
"Geometric Incidence Theorems via Fourier Analysis,"
Transactions of the American Mathematical Society, 2009
We show that every non-trivial Sobolev bound for generalized Radon transforms which average functions over families of curves and surfaces yields an incidence theorem for suitably regular discrete sets of points and curves or surfaces in Euclidean space. This mechanism allows us to deduce geometric results not readily accessible by combinatorial methods.
Average decay of Fourier transforms and geometry of convex sets
Revista Matemática Iberoamericana, 2000
Let B be a convex body in R 2 , with piecewise smooth boundary and let b B denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical L p -averages of b B and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average numberofinteger lattice points in large bodies randomly positioned in the plane.