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## Über dieses Buch

Asymptotic Geometric Analysis is concerned with the geometric and linear properties of finite dimensional objects, normed spaces, and convex bodies, especially with the asymptotics of their various quantitative parameters as the dimension tends to infinity. The deep geometric, probabilistic, and combinatorial methods developed here are used outside the field in many areas of mathematics and mathematical sciences. The Fields Institute Thematic Program in the Fall of 2010 continued an established tradition of previous large-scale programs devoted to the same general research direction. The main directions of the program included:

* Asymptotic theory of convexity and normed spaces

* Concentration of measure and isoperimetric inequalities, optimal transportation approach

* Applications of the concept of concentration

* Connections with transformation groups and Ramsey theory

* Geometrization of probability

* Random matrices

* Connection with asymptotic combinatorics and complexity theory

These directions are represented in this volume and reflect the present state of this important area of research. It will be of benefit to researchers working in a wide range of mathematical sciences—in particular functional analysis, combinatorics, convex geometry, dynamical systems, operator algebras, and computer science.

## Inhaltsverzeichnis

### The Variance Conjecture on Some Polytopes

We show that any random vector uniformly distributed on any hyperplane projection of

B

1

n

or

B

n

verifies the variance conjecture

$$\text{Var }\vert X{\vert }^{2} \leq C\sup\limits_{ \xi \in {S}^{n-1}}\mathbb{E}\langle X,{\xi \rangle }^{2}\mathbb{E}\vert X{\vert }^{2}.$$

Furthermore, a random vector uniformly distributed on a hyperplane projection of

B

n

verifies a negative square correlation property and consequently any of its linear images verifies the variance conjecture.

David Alonso–Gutiérrez, Jesús Bastero

### More Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures

In this paper, we compute universal minimal flows of groups of automorphisms of uncountable ω-homogeneous graphs,

K

n

-free graphs, hypergraphs, partially ordered sets, and their extensions with an ω-homogeneous ordering. We present an easy construction of such structures, expanding the jungle of extremely amenable groups.

Dana Bartošová

### On the Lyapounov Exponents of Schrödinger Operators Associated with the Standard Map

It is shown that Schrödinger operators defined from the standard map have positive (mean) Lyapounov exponents for almost all energies.

J. Bourgain

### Overgroups of the Automorphism Group of the Rado Graph

We are interested in overgroups of the automorphism group of the Rado graph. One class of such overgroups is completely understood; this is the class of reducts. In this article we tie recent work on various other natural overgroups, in particular establishing group connections between them and the reducts.

Peter Cameron, Claude Laflamme, Maurice Pouzet, Sam Tarzi, Robert Woodrow

### On a Stability Property of the Generalized Spherical Radon Transform

In this note, we study the operator norm of the generalized spherical Radon transform, defined by a smooth measure on the underlying incidence variety. In particular, we prove that for small perturbations of the measure, the spherical Radon transform remains an isomorphism between the corresponding Sobolev spaces.

Dmitry Faifman

### Banach Representations and Affine Compactifications of Dynamical Systems

To every Banach space

V

we associate a compact right topological affine semigroup

(

V

). We show that a separable Banach space

V

is Asplund if and only if

$$\mathcal{E}(V )$$

is metrizable, and it is Rosenthal (i.e., it does not contain an isomorphic copy of

l

1

) if and only if

$$\mathcal{E}(V )$$

is a Rosenthal compactum. We study representations of compact right topological semigroups in

$$\mathcal{E}(V )$$

. In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily nonsensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system

X

and the enveloping semigroup of its various affine compactifications

Q

(

X

). When the two coincide we say that the affine compactification

Q

(

X

) is

E

-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any

E

-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and

E

-compatibility.

Eli Glasner, Michael Megrelishvili

### Flag Measures for Convex Bodies

Measures on flag manifolds have been recently used to describe local properties of convex bodies and more general sets in

$${\mathbb{R}}^{d}$$

. Here, we provide a systematic account of flag measures for convex bodies, we collect various properties of flag measures and we prove some new results. In particular, we discuss mixed flag measures for several bodies and we present formulas for (mixed) flag measures of generalized zonoids.

Daniel Hug, Ines Türk, Wolfgang Weil

### Operator Functional Equations in Analysis

Classical operations in analysis and geometry as derivatives, the Fourier transform, the Legendre transform, multiplicative maps or duality of convex bodies may be characterized, essentially, by very simple properties which may be often expressed as operator equations, like the Leibniz or the chain rule, bijective maps exchanging products with convolutions or bijective order reversing maps on convex functions or convex bodies. We survey and discuss recent results of this type in analysis. The operations we consider act on classical spaces like

C

k

-spaces or Schwartz spaces

$$\mathcal{S}({\mathbb{R}}^{n})$$

. Naturally, the results strongly depend on the type of the domain and the image space.

Hermann König, Vitali Milman

### A Remark on the Extremal Non-Central Sections of the Unit Cube

In this paper, we investigate extremal volumes of non-central slices of the unit cube. The case of central hyperplane sections is known and was studied by Ball, Hadwiger and Hensley. The case of non-central sections, i.e. when we dictate that the hyperplane must be a certain distance

t

> 0 from the center of the cube, is open in general and the same is true about sections of the unit cube by slabs. In this paper we give a full solution for extremal one-dimensional sections and a partial solution for extremal hyperplane slices for the case

$$t > \frac{\sqrt{n-1}} {2}$$

. We also make a remark on minimal volume slices of the cube by slabs of width 2

t

, when

$$t > \frac{\sqrt{n-1}} {2}$$

.

James Moody, Corey Stone, David Zach, Artem Zvavitch

### Universal Flows of Closed Subgroups of S ∞ and Relative Extreme Amenability

This paper is devoted to the study of universality for a particular continuous action naturally attached to certain pairs of closed subgroups of

S

. It shows that three new concepts, respectively called relative extreme amenability, relative Ramsey property for embeddings and relative Ramsey property for structures, are relevant in order to understand this property correctly. It also allows us to provide a partial answer to a question posed in [2] by Kechris, Pestov and Todorcevic (Geom. Funct. Anal.

15

(1), 106–189, 2005).

L. Nguyen Van Thé

### Oscillation of Urysohn Type Spaces

A metric space

$$\mathrm{M} = (M;d )$$

is

homogeneous

if for every isometry

$$\alpha$$

of a finite subspace of

$$\mathrm{M}$$

to a subspace of

$$\mathrm{M}$$

there exists an isometry of

$$\mathrm{M}$$

onto

$$\mathrm{M}$$

extending

$$\alpha$$

. The metric space

$$\mathrm{M}$$

is

universal

if it isometrically embeds every finite metric space

$$\mathrm{F}$$

with

$$dist (\mathrm{F}) \subseteq dist= (\mathrm{M})$$

. (

$$dist (\mathrm{M})$$

being the set of distances between points of

$$\mathrm{M}$$

.) A metric space

$$\mathrm{M}$$

is

oscillation stable

if for every

$$\epsilon > 0$$

and every uniformly continuous and bounded function

$$f : M \rightarrow \mathfrak{R}$$

there exists an isometric copy

$$\mathrm{{M}}^{{\ast}} = ({M}^{{\ast}};d )$$

of

$$\mathrm{M}$$

in

$$\mathrm{M}$$

for which:

$$\sup \{\vert f(x) - f(y)\vert \mid x,y \in {M}^{{\ast}}\} < \epsilon.$$

Theorem.

Every bounded, uncountable, separable, complete, homogeneous, universal metric space

$$\mathrm{M} = (M;d )$$

is oscillation stable. (Theorem 12.)

N. W. Sauer

### Euclidean Sections of Convex Bodies

This is a somewhat expanded form of a 4h course given, with small variations, first at the educational workshop Probabilistic methods in geometry, Bedlewo, Poland, July 6–12, 2008 and a few weeks later at the Summer school on Fourier analytic and probabilistic methods in geometric functional analysis and convexity, Kent, Ohio, August 13–20, 2008. The main part of these notes gives yet another exposition of Dvoretzky’s theorem on Euclidean sections of convex bodies with a proof based on Milman’s. This material is by now quite standard. Towards the end of these notes we discuss issues related to fine estimates in Dvoretzky’s theorem and there are some results that didn’t appear in print before. In particular there is an exposition of an unpublished result of Figiel (Claim1) which gives an upper bound on the possible dependence on

$$\epsilon$$

in Milman’s theorem. We would like to thank Tadek Figiel for allowing us to include it here. There is also a better version of the proof of one of the results from Schechtman (Adv. Math.

200

(1), 125–135, 2006) giving a lower bound on the dependence on

$$\epsilon$$

in Dvoretzky’s theorem. The improvement is in the statement and proof of Proposition 2 here which is a stronger version of the corresponding Corollary 1 in Schechtman (Adv. Math.

200

(1), 125–135, 2006).

Gideon Schechtman

### Duality on Convex Sets in Generalized Regions

Recently, the duality relation on several families of convex sets was shown to be completely characterized by the simple property of reversing order. The families discussed in aforementioned results were convex sets in

$${\mathbb{R}}^{n}$$

. Our goal in this note is to generalize this type of results to regions in

$${\mathbb{R}}^{n}$$

bounded between two convex sets.

Alexander Segal, Boaz A. Slomka

### On Polygons and Injective Mappings of the Plane

We give an affirmative answer to a question asked by Gardner and Mauldin (Geom. Dedicata

26

, 323–332, 1988) about bijections of the plane taking each polygon with

n

sides onto a polygon with

n

sides. We also state and prove more general results in this spirit. For example, we show that an injective mapping taking each convex

n

-gon onto a non-degenerate

n

-gon (not necessarily convex or even simple) must be affine.

Boaz A. Slomka

### Abstract Approach to Ramsey Theory and Ramsey Theorems for Finite Trees

I will give a presentation of an abstract approach to finite Ramsey theory found in an earlier paper of mine. I will prove from it a common generalization of Deuber’s Ramsey theorem for regular trees and a recent Ramsey theorem of Jasiński for boron tree structures. This generalization appears to be new. I will also show, in exercises, how to deduce from it the Milliken Ramsey theorem for strong subtrees.

Sławomir Solecki

### Some Affine Invariants Revisited

We present several sharp inequalities for the

SL

(

n

) invariant

Ω

2,

n

(

K

) introduced in our earlier work on centro-affine invariants for smooth convex bodies containing the origin. A connection arose with the Paouris-Werner invariant

Ω

K

defined for convex bodies

K

whose centroid is at the origin. We offer two alternative definitions for

Ω

K

when

K

C

+

2

. The technique employed prompts us to conjecture that any

SL

(

n

) invariant of convex bodies with continuous and positive centro-affine curvature function can be obtained as a limit of normalized

p

-affine surface areas of the convex body.

Alina Stancu

### On the Geometry of Log-Concave Probability Measures with Bounded Log-Sobolev Constant

Let

$$\mathcal{L}S_{lc}(\kappa )$$

denote the class of log-concave probability measures

μ

on

$${\mathbb{R}}^{n}$$

which satisfy the logarithmic Sobolev inequality with a given constant κ > 0. We discuss

$$\mathcal{L}S_{lc}(\kappa )$$

from a geometric point of view and we focus on related open questions.

P. Stavrakakis, P. Valettas

### f-Divergence for Convex Bodies

We introduce

f

-divergence, a concept from information theory and statistics, for convex bodies in ℝ

n

. We prove that

f

-divergences are

SL

(

n

) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the

L

p

affine surface area from the

L

p

Brunn Minkowski theory are special cases of

f

-divergences.

Elisabeth M. Werner
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