Alexander Koldobsky

Fourier Analysis in Convex Geometry

Introduction Basic concepts Volume and the Fourier transform Intersection bodies The Busemann-Petty problem Intersection bodies and

An analytic solution to the Busemann-Petty problem on sections of convex bodies

L_p

A functional analytic approach to intersection bodies

-spaces Extremal sections of

A generalization of the Busemann-Petty problem on sections of convex bodies

\ell_q

An application of the fourier transform to sections of star bodies

-balls Projections and the Fourier transform Bibliography Index.

Morphological regularization neural networks

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 (ni 1)dimensional X-ray) gives the ((ni 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 unifled analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies inR n such that the ((ni 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 deflnite distributions, our formula shows that the answer to the problem depends on the behavior of the (ni 2)-nd derivative of the parallel section functions. The a‐rmative 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.

On the Central Limit Property of Convex Bodies

Abstract. We consider several generalizations of the concept of an intersection body and show their connections with the Fourier transform and embeddings in Lp-spaces. These connections lead to generalizations of the recent solution to the Busemann—Petty problem on sections of convex bodies.

Operators preserving orthogonality are isometries

The 1956 Busemann-Petty problem asks whether symmetric convex bodies in ℝn with larger central hyperplane sections must also have greater volume. The solution to the problem has recently been completed, and the answer is negative ifn≥5 and affirmative whenn≤4. We show a more general result, where the inequalities for the volume of central sections are replaced by similar inequalities for the derivatives of the parallel section functions at zero. The dimension of affirmative answer goes up together with the order of the derivatives. The proof is based on a version of Parsevals formula.

THE COMPLEX BUSEMANN-PETTY PROBLEM ON SECTIONS OF CONVEX BODIES

We express the volume of central hyperplane sections of star bodies inRn in terms of the Fourier transform of a power of the radial function, and apply this result to confirm the conjecture of Meyer and Pajor on the minimal volume of central sections of the unit balls of the spacesℓpn with 0

The Geometry of

Abstract In this paper we establish a relationship between regularization theory and morphological shared-weight neural networks (MSNN). We show that a certain class of morphological shared-weight neural networks with no hidden units can be viewed as regularization neural networks. This relationship is established by showing that this class of MSNNs are solutions of regularization problems. This requires deriving the Fourier transforms of the min and max operators. The Fourier transforms of min and max operators are derived using generalized functions because they are only defined in that sense.