Projections of convex bodies and the fourier transform

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 ofl _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 ofl _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.     

On a spherical integral transformation and sections of star bodies

LetK be ad-dimensional star body (with respect to the origino). It is known that the (d–1)-dimensional volume of the intersections ofK with the hyperplanes througho does not uniquely determineK. Uniqueness can only be achieved under additional assumptions, such as central symmetry. Here it is pointed out that if one uses, instead of intersections by hyperplanes, intersections by half-planes that containo on the boundary, then, without any additional assumptions, the volume of these intersections determinesK uniquely. This assertion, and more general results of this kind, together with stability estimates, are obtained from uniqueness results and estimates concerning a particular spherical integral transformation.Supported by National Science Foundation Research Grant DMS-9401487     

Geometry of the finite fourier transform

We provide a geometric interpretation for the multidimensional finite Fourier transform and give a reformulation of the Macwilliams identitity in terms of theta functions.     

On vectorizing the fast fourier transform

A variant of the Cooley-Tukey algorithm due to Stockham is derived and vectorized and is shown to be on a par with the Pease algorithm. The Stockham algorithm is then proposed for the entire computation of the two-dimensional fast Fourier transform on a vector computer.     

Matrix identities of the fast fourier transform

Let A_n(ω) be the nxn matrix A_n(ω)=(a_ij with a_ij=ω^ij, 0?i,j?n?1, ωⁿ=1. For n=rs we show

$\text{A}{}_{\mathrm{n}}\text{(w)P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{P}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{∵}\text{0}\text{s?1}\text{A}{}_{\mathrm{r}}\text{(w}{}^{\mathrm{s}}\text{)}\text{P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{{T}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{(w)}}\text{∵}\text{0}\text{r?1}\text{A}{}_{\mathrm{s}}\text{(w}{}^{\mathrm{r}}\text{)}$

=(A_r?I_s)T^s_r(I_r?A_s). When r and s are relatively prime this identity implies a wide class of identities of the form PA_n(ω)Q^T=A_r(ω^αs)?A_s(ω^βr). The matrices P^s_r, P^r_s, P, and Q are permutation matrices corresponding to the “data shuffling” required in a computer implementation of the FFT, and T^s_r is a diagonal matrix whose nonzeros are called “twiddle factors.” We establish these identities and discuss their algorithmic significance.     

Determination of star bodies from p-centroid bodies

In this paper, we prove that an origin-symmetric star body is uniquely determined by its p-centroid body. Furthermore, using spherical harmonics, we establish a result for non-symmetric star bodies. As an application, we show that there is a unique member of $\Gamma_p\langle K\rangle$ characterized by having larger volume than any other member, for all real p?≥?1 that are not even natural numbers, where $\Gamma_p\langle K\rangle$ denotes the p-centroid equivalence class of the star body K.     

Generalized fourier transform of slow-growth distributions

Krasnoyarsk City. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 2, pp. 94–103, March–April, 1990.     

The arithmetic fourier transform and the visual pathway

The Arithmetic Fourier Transform is an algorithm for the computation of Fourier coefficients. In this paper it is extended to the computation of double Fourier coefficients by a repeated sum. We discuss the relevance of the repeated sum algorithm to signal processing by neurons in the visual pathway.     

Free channel fourier transform in the long-rangen-body problem

Research supported by NSF grant DMS-8807816.     

Non-central sections of convex bodies

Let K be a convex body in ? ⁿ . Is K uniquely determined by the areas of its sections? There are classical results that explain what happens in the case of sections passing through the origin. However, much less is known about sections that do not contain the origin. We discuss several problems of this type and establish the corresponding uniqueness results.     

Hyperplane sections of convex bodies

We prove sharp inequalities for the volumes of hyperplane sections bisecting a convex body in Rn. This leads to a relative isoperimetric inequality for arbitrary hyperplane sections of a convex body.     

An application of the principle of limiting absorption to the motions of floating bodies

We show a uniqueness and existence theorem for the so-called linearized seakeeping problem with a range of application much wider than that of the previous results of F. John (Comm. Pure Appl. Math.3 (1950), 45–101).     

Estimates of the asphericity of sections of convex bodies

We define the function (n, k) to be the infimum of all such that any bounded centrally symmetric convex body inR ⁿ possesses an -asphericalk-dimensional central section. It is proved that (3, 2)=2–1 and (n, n-1)n-1-1. Several related functions are defined and their values on the pairs (n, n-1) are estimated.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 76–79.