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Concentration of Measure for the Analysis of Randomized Algorithms by Alessandro Panconesi, Devdatt P. Dubhashi

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11

Talagrand’s Isoperimetric Inequality

In this chapter we introduce a powerful and versatile inequality due to Talagrand [85]. Our aim here is to state the inequality without proof, derive nicely packaged versions of it, as we did in the case of martingales and the method of bounded differences, and illustrate them with applications to several examples. We defer the proof to later chapters: in Chapters 13 and 14 we give two different proofs after developing the necessary machinery. Expositions of direct proofs appear in [2, 67, 84] in addition to the original [85].

11.1 Statement of the Inequality

Talagrand’s inequality is an isoperimetric inequality where the familiar notion of a Hamming metric is replaced by a more general notion of Talagrand’s ...

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