Apostolos Giannopoulos

Geometry of isotropic convex bodies

Background from asymptotic convex geometry Isotropic log-concave measures Hyperplane conjecture and Bourgains upper bound Partial answers L q -centroid bodies and concentration of mass Bodies with maximal isotropic constant Logarithmic Laplace transform and the isomorphic slicing problem Tail estimates for linear functionals M and M? *-estimates Approximating the covariance matrix Random polytopes in isotropic convex bodies Central limit problem and the thin shell conjecture The thin shell estimate Kannan-Lov sz-Simonovits conjecture Infimum convolution inequalities and concentration Information theory and the hyperplane conjecture Bibliography Subject index Author index

Extremal problems and isotropic positions of convex bodies

LetK be a convex body in ℝn and letWi(K),i=1, …,n−1 be its quermassintegrals. We study minimization problems of the form min{Wi(TK)|T ∈ SLn} and show that bodies which appear as solutions of such problems satisfy isotropic conditions or even admit an isotropic characterization for appropriate measures. This shows that several well known positions of convex bodies which play an important role in the local theory may be described in terms of classical convexity as isotropic ones. We provide new applications of this point of view for the minimal mean width position.

A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies

Let It is proved that for suitable a and b , n ≥7, one can have V n ( A n ) = V n ( B n ) and for every ( n –1)-dimensional subspace H of ℝ n , where B n is the unit ball of ℝ n . This strengthens previous negative results on a problem of H. Busemann and C. M. Petty.

Chapter 17 – Euclidean Structure in Finite Dimensional Normed Spaces

This chapter discusses the results that stand between geometry, convex geometry, and functional analysis. The chapter describes the family of n -dimensional normed spaces and discusses the study on the asymptotic behavior of their parameters, as the dimension n grows to infinity. This theory grew out of functional analysis. In fact, it may be viewed as the most recent one among many examples of directions in mathematics that were born inside this field during the twentieth century. The influence of the ideas of functional analysis on mathematical physics not only on differential equations but also on classical analysis was enormous. The great achievements and successful applications to other fields led to the creation of new directions (among them, algebraic analysis, noncommutative geometry, and the modem theory of partial differential equations) that, in a short time, became autonomous and independent fields of mathematics.

John's Theorem for an Arbitrary Pair of Convex Bodies

We provide a generalization of Johns representation of the identity for the maximal volume position of L inside K, where K and L are arbitrary smooth convex bodies in ℝn. From this representation we obtain Banach–Mazur distance and volume ratio estimates.

On the mean value of the area of a random polygon in a plane convex body

Let K be a convex body in Euclidean space R d , d ≥2, with volume V ( K ) = 1, and n ≥ d +1 be a natural number. We select n independent random points y 1 , y 2 , …, y n from K (we assume they all have the uniform distribution in K ). Their convex hull co { y 1 , y 2 , …, y n } is a random polytope in K with at most n vertices. Consider the expected value of the volume of this polytope It is easy to see that if U : R d → R d is a volume preserving affine transformation, then for every convex body K with V ( K ) = 1, m ( K, n ) = m ( U ( K ), n ).

A NOTE ON THE BANACH-MAZUR DISTANCE TO THE CUBE

If X is an n-dimensional normed space, and d denotes the Banach-Mazur distance, then d(X, l ∞ n ) ≤ cn 5/6.

Random Points in Isotropic Unconditional Convex Bodies

The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies

Asymptotic Convex Geometry Short Overview

K, T_1,\ldots, T_s

Some inequalities about mixed volumes

. (a) Let