Γεωμετρία των ισοτροπικών λογαριθμικά-κοίλων μέτρων

Abstract

The results presented in this thesis belong to the theory of isotropic convex bodies or, moregenerally, isotropic logarithmically-concave measures. One very well-known problem in this theoryis the slicing problem or isotropic constant problem, which asks whether the isotropic constants ofall convex bodies contained in a nite-dimensional Euclidean space, or more generally the isotropicconstants of nite log-concave measures dened on a nite-dimensional space, can be bounded fromabove by a constant independent of the dimension of the space.In Chapters 2 and 3 of this thesis we present two dierent reductions of the slicing problem.The rst, which is joint work with Giannopoulos and Paouris [4], is a reduction of the problemto the study of the behaviour of the rst moment with respect to the uniform measure on someisotropic convex body K of the support function of the centroid bodies of K. We show that even asmall improvement to the currently known upper bounds for these moments would lead to improvedupper bounds for the isotropic constants of log-concave measures. The behaviour and distributionof the support function of the centroid bodies of an isotropic convex body K contained in a nite-dimensional Euclidean space is directly related to the behaviour of the linear functionals denedon the space and their distribution in the convex body K. This behaviour is a crucial ingredient ofother approaches to the isotropic constant problem as well (see e.g. [1], [3]). Our reduction can beseen as a continuation to these approaches.In Chapter 3 we compare and combine two recent approaches to the isotropic constant problemthat were introduced by Dafnis and Paouris [3] and by Klartag and E. Milman [6]. The methodof Dafnis and Paouris reduces the slicing problem to the study of the negative moments of theEuclidean norm with respect to some isotropic log-concave measure, while the method of Klartagand E. Milman leads to lower bounds for the volume of the centroid bodies of an isotropic log-concave measure, from which we can also infer upper bounds for the isotropic constant of themeasure. In Chapter 3 we dene two new parameters for every isotropic log-concave measure andwe show how the method of Klartag and E. Milman, and the apparently stronger conclusions itleads to, can be extended in the full range of the \weaker" assumptions of Dafnis and Paouris.As a consequence, we get a second reduction of the slicing problem that was introduced in [7].The new parameters that we dene for every isotropic log-concave measure on an n-dimensionalEuclidean space are related to the highest dimension k < n of subspaces in which we can ndmarginal measures of with absolutely bounded isotropic constant.Finally, in Chapter 4 we present a purely geometric proof of the reverse Santalo inequality. Thisproof is joint work with Giannopoulos and Paouris [5]. Recall that according to the reverse Santaloinequality, or the Bourgain-Milman inequality [2], for every convex body K in an Euclidean spaceof dimension n that contains the origin in its interior, the n-th root of the product of the volume ofK and the volume of its polar is at least of the order 1/n. It is known that it suces to show theinequality for convex bodies K with centre of mass at the origin, and also that the above volumeproduct is invariant by the action of invertible linear transformations. As a consequence, for theproof that we describe in Chapter 4 it is possible to assume at rst that K is in isotropic position,which allows us to combine well-known and relatively easy to prove properties of isotropic convexbodies to derive the reverse Santalo inequality.