| Abstract |
Let $(X,{\mathcal A},\mu )$ be a probability space, where ${\mathcal A}$ is the Borel $\sigma $-algebra with respect to a given metric $d$ on $X$. We say that the quadratuple $(X,{\mathcal A},\mu ,d)$ is a metric probability space. For any metric probability space, we may state the isoperimetric problem: Given $0<\alpha <1$ and $t>0$, find the infimum $$\inf\{\mu (A_t):A\in {\mathcal A},\mu (A)=\alpha\}$$ and the sets $A$ for which it is attained. In the question above, we denote by $A_t$ the $t$-extension of $A$: $$A_t=\{ x\in X:d(x,A)\leq t\}.$$ In the first Chapter of this work we discuss some classical cases of isoperimetric problems for which the precise answer is known. There are many other isoperimetric problems with important applications, for which a precise answer is impossible. However, a weaker answer is equally useful: instead of finding the precise value of the infimum, it is enough to know a good lower bound for $\mu (A_t)$ under the assumption that $\mu (A)=\alpha $. We will say that such a bound solves the isoperimetric problem ``approximately" if it provides an optimal estimate with the exception of some absolute constants in the ``right place". Inequalities from which such approximate solutions follow, are called approximate isoperimetric inequalities. In this work we describe several methods of proof of approximate isoperimetric inequalities for products of general metric probability spaces. The tools are geometric, combinatorial and probabilistic. We present the following techniques: 1. Talagrand's inductive method for abstract product measures. We study several notions of distance leading to different notions of extension of a set. 2. The technique of martingales. An example of application is concentration of measure in the space of permutations. 3. Property $(\tau )$ (Maurey's method of the infimum convolution). 4. Logarithmic Sobolev inequality and the concentration of measure in Gauss space.
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