2,687 research outputs found
Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities
We prove a family of uncertainty inequalities on fairly general groups
and homogeneous spaces, both in the smooth and in the discrete setting. The
crucial point is the proof of the endpoint, which is derived from a
general weak isoperimetric inequality.Comment: 17 page
Extremal Lipschitz functions in the deviation inequalities from the mean
We obtain an optimal deviation from the mean upper bound \begin{equation}
D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\
x\in\R\label{abstr} \end{equation} where \F is the class of the integrable,
Lipschitz functions on probability metric (product) spaces. As corollaries we
get exact solutions of \eqref{abstr} for Euclidean unit sphere with
a geodesic distance and a normalized Haar measure, for 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 in \eqref{abstr} is achieved on
a family of distance functions.Comment: 7 page
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