The Busemann-Petty problem via spherical harmonics

The Busemann-Petty problem asks whether symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller n-dimensional volume. The solution has recently been completed, and the answer is affirmative if n?4 and negative if n?5. In this article we present a short proof of the affirmative result and its generalization using the Funk-Hecke formula for spherical harmonics.     

The Busemann-Petty problem for arbitrary measures

The Busemann-Petty problem asks whether symmetric convex bodies in ⁿ with smaller (n–1)-dimensional volume of central hyperplane sections necessarily have smaller n-dimensional volume. The answer to this problem is affirmative for n4 and negative for n5. In this paper we generalize the Busemann-Petty problem to essentially arbitrary measure in place of the volume. We also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in ⁿ.Mathematics Subject Classification (2000): 52A15, 52A21, 52A38     

The complex Busemann-Petty problem on sections of convex bodies

The complex Busemann-Petty problem asks whether origin symmetric convex bodies in Cn with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if n?3 and negative if n?4.     

On the generalized Busemann-Petty problem

The generalized Busemann-Petty problem asks whether the origin-symmetric convex bodies in ℝ ⁿ with a larger volume of all i-dimensional sections necessarily have a larger volume. As proved by Bourgain and Zhang, the answer to this question is negative if i > 3. The problem is still open for i = 2, 3. In this article we prove two specific affirmative answers to the generalized Busemann-Petty problem if the body with a smaller i-dimensional volume belongs to given classes. Our results generalize Zhang’s specific affirmative answer to the generalized Busemann-Petty problem. This work was supported, in part, by the National Natural Science Foundation of China (Grant No. 10671117)     

The complex Busemann-Petty problem for arbitrary measures

The complex Busemann-Petty problem asks whether origin symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if and negative if . In this article we show that the answer remains the same if the volume is replaced by an “almost” arbitrary measure. This result is the complex analogue of Zvavitch’s generalization to arbitrary measures of the original real Busemann-Petty problem. Received: 6 May 2008     

Modified Busemann-Petty problem on sections of convex bodies

The Busemann-Petty problem asks whether convex origin-symmetric bodies in ℝ ⁿ with smaller central hyperplane sections necessarily have smallern-dimensional volume. It is known that the answer is affirmative ifn≤4 and negative ifn≥5. In this article we replace the assumptions of the original Busemann-Petty problem by certain conditions on the volumes of central hyperplane sections so that the answer becomes affirmative in all dimensions. The first-named author was supported in part by the NSF grant DMS-0136022 and by a grant from the University of Missouri Research Board.     

The modified complex Busemann-Petty problem on sections of convex bodies

The complex Busemann-Petty problem asks whether origin symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n ≤ 3 and negative if n ≥ 4. Since the answer is negative in most dimensions, it is natural to ask what conditions on the (n − 1)-dimensional volumes of the central sections of complex convex bodies with complex hyperplanes allow to compare the n-dimensional volumes. In this article we give necessary conditions on the section function in order to obtain an affirmative answer in all dimensions. The result is the complex analogue of [16].      

The Busemann-Petty problem in hyperbolic and spherical spaces

The Busemann-Petty problem asks whether origin-symmetric convex bodies in Rⁿ with smaller central hyperplane sections necessarily have smaller n-dimensional volume. It is known that the answer to this problem is affirmative if n?4 and negative if n?5. We study this problem in hyperbolic and spherical spaces.     

On an analytic generalization of the Busemann-Petty problem

In this paper, we establish an extension of the connections between an analytic generalization of the Busemann-Petty problem and the positive definite distributions. Our results show that the structure of the positive definite distributions in Rn is closely related to the analytic generalization of the Busemann-Petty problem which was posed by Koldobsky.     

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

It is proved that for arbitrarymεℕ and for a sufficiently nontrivial compact groupG of operators acting on a “typical”n-dimensional quotientX _n ofl ₁ ^m withm=(1+δ)n, there is a constantc=c(δ) such that Supported in part by KBN grant no. 2 P03A 034 10.     

The first variation of the total mass of log-concave functions and related inequalities

On the class of log-concave functions on Rⁿ${\mathbb{R}}^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such a first variation, which suggest to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski’s first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski’s problem for convex bodies, and prove some partial results towards its solution.     

The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry

The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. If i > 3, the answer is negative. For i = 2 or i = 3 (n > 4), the problem is still open, however, when the body with smaller sections is a body of revolution, the answer is affirmative. The paper contains a solution to the problem in the more general situation, when the body with smaller sections is invariant under rotations, preserving mutually orthogonal subspaces of dimensions ℓ and n − ℓ, respectively, so that i + ℓ ≤ n. The answer essentially depends on ℓ. The argument relies on the notion of canonical angles between subspaces, spherical Radon transforms, properties of intersection bodies, and the generalized cosine transforms.     

Generalizations of the Busemann-Petty problem for sections of convex bodies

We present generalizations of the Busemann-Petty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For two- three-dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions.     

Possible entropy functions

In this article, we consider certain systems of derivation algebras related to Galois representations attached to fundamental groups of algebraic curves of positive genera and establish some stability property. This is a generalization of Ihara’s result in the case of genus zero. This research was partly supported by National Science Foundation Grant 09740041 and 12740026.     

A solution to the lower dimensional Busemann-Petty problem in the hyperbolic space

The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in ℝⁿ with smaller volume of all k-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if k>3. The problem is still open for k = 2, 3. In this article we formulate and completely solve the lower dimensional Busemann-Petty problem in the hyperbolic space ℍⁿ.     

On directional entropy functions

Given aZ ²-process, the measure theoretic directional entropy function,h( % MathType!End!2!1!), is defined on % MathType!End!2!1!. We relate the directional entropy of aZ ²-process to itsR ² suspension. We find a sufficient condition for the continuity of directional entropy function. In particular, this shows that the directional entropy is continuous for aZ ²-action generated by a cellular automaton; this finally answers a question of Milnor [Mil]. We show that the unit vectors whose directional entropy is zero form aG _δ subset ofS ¹. We study examples to investigate some properties of directional entropy functions. This research is supported in part by BSRI and KOSEF 95-0701-03-3.     

A note on entropy and inner functions

This paper constructs an inner function with infinite entropy.     

Topological entropy of continuous functions on topological spaces

Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen’s entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew’s entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew’s entropy for compact systems.     

The American put is log-concave in the log-price

We show that the American put option price is log-concave as a function of the log-price of the underlying asset. Thus the elasticity of the price decreases with increasing stock value. We also consider related contracts of American type, and we provide an example showing that not all American option prices are log-concave in the stock log-price.