Geometric inequalities and advances in the Ribe program

Abstract

In Chapter 1 we show that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature, thus answering a question of Gromov (1993). In Chapter 2 we introduce a metric invariant called diamond convexity and show that a Banach space $(X,\|\cdot\|_X)$ has diamond convexity $q$ if and only if $X$ admits an equivalent $q$-uniformly convex norm. We also study the relation of diamond convexity with other metric invariants, such as Markov convexity and metric cotype. In Chapter 3 we use metric invariants introduced in the context of the Ribe program to derive nonembeddability results for subsets of $L_p$ spaces. In Chapter 4 we prove that the dependence on the dimension in Pisier's inequality for superreflexive targets $(X,\|\cdot\|_X)$ is $O\big((\log n)^{\alpha(X)}\big)$ for some $\alpha(X)\in[0,1)$, thus providing the first improvement of Pisier's original $O(\log n)$ bound (1986) for this class of spaces. In Chapter 5 we undertake a systematic investigation of dimension independent properties of vector valued functions with bounded spectrum defined on the discrete cube. Our approach relies on input from discrete harmonic analysis, complex analysis and classical approximation theory. In Chapter 6 we use the method of Chapter 5 to derive the first dimension independent extension of Freud's inequality (1971). In Chapter 7 we introduce a probabilistic technique relying on mixtures of Gaussian measures to address problems in information theory and convex geometry, for instance proving extensions of the B-inequality and the Gaussian correlation inequality. In Chapter 8 we use a different probabilistic approach to derive the sharp constants in the Khintchine inequality for vectors uniformly distributed on the unit ball of $\ell_p^n$, thus answering a question of Barthe, Gu\'edon, Mendelson and Naor (2005), and solve a variant of the classical moment problem for symmetric log-concave distributions on the real line. In Chapter 9 we identify the extremal block hyperplane sections of spaces of the form $\ell_p^n(X)$, where $p\in(0,2]$ and $X$ is a quasi-Banach space which admits an isometric embedding into $L_p$. In Chapter 10 we show that the $(n,k)$ Bernoulli--Laplace urn model exhibits mixing time cutoff after $\frac{n}{4k}\log n$ steps when $k=o(n)$.