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Extremal Lipschitz functions in the deviation inequalities from the mean

   
@article{ECP2814,
	author = {Dainius Dzindzalieta},
	title = {Extremal Lipschitz functions in the deviation inequalities from the mean},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {18},
	year = {2013},
	keywords = {Gaussian, vertex isoperimetric, deviation from the mean, inequalities, Hamming, probability metric space},
	abstract = {We obtain an optimal deviation from the mean upper bound $D(x)=\sup\{\mu\{f-\mathbb{E}_{\mu} f\geq x\}:f\in\mathcal{F},x\in\mathbb{R}\}$ where $\mathcal{F}$ is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact bounds for Euclidean unit sphere $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\mathbb{R}^n$ equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance. We also prove that in general probability metric spaces the $\sup$ is achieved on a family of distance functions.},
	pages = {no. 66, 1-5},
	issn = {1083-589X},
	doi = {10.1214/ECP.v18-2814},    
        url = {http://ecp.ejpecp.org/article/view/2814}}