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

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 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<

On the mean-width of isotropic convex bodies and their associated LpL_p-centroid bodies

For any origin-symmetric convex body $K$ in $\mathbb{R}^n$ in isotropic position, we obtain the bound: $$M^*(K) \leq C \sqrt{n} \log(n)^2 L_K ~,$$ where $M^*(K)$ denotes (half) the mean-width of $K$, $L_K$ is the isotropic constant of $K$, and $C>0$ is a universal constant. This improves the previous best-known estimate $M^*(K) \leq C n^{3/4} L_K$. Up to the power of the $\log(n)$ term and the $L_K$ one, the improved bound is best possible, and implies that the isotropic position is (up to the $L_K$ term) an almost $2$-regular $M$-position. The bound extends to any arbitrary position, depending on a certain weighted average of the eigenvalues of the covariance matrix. Furthermore, the bound applies to the mean-width of $L_p$-centroid bodies, extending a sharp upper bound of Paouris for $1 \leq p \leq \sqrt{n}$ to an almost-sharp bound for an arbitrary $p \geq \sqrt{n}$. The question of whether it is possible to remove the $L_K$ term from the new bound is essentially equivalent to the Slicing Problem, to within logarithmic factors in $n$.Comment: 15 pages; added references, to appear in IMRN. See publisher's website for final versio

Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry--\'Emery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension \emph{independent} bounds. As a corollary, we can recover and extend all previously known (dimension dependent) results by generalizing an isoperimetric inequality of Bobkov, and provide a new proof that under natural convexity assumptions, arbitrarily weak concentration implies a dimension independent linear isoperimetric inequality. Further applications will be described in a subsequent work. Contrary to previous attempts in this direction, our method is entirely geometric, continuing the approach set forth by Gromov and adapted to the manifold-with-density setting by Morgan.Comment: 28 pages; to appear in Duke Math. J. - shortened exposition and addressed referees' useful comment