An Analysis Upon some Remarks on Gaussian Isoperimetric Inequality: a New Approach |
The Gaussian isoperimetric inequality, and its relatedconcentration phenomenon, is one of the most important properties of Gaussianmeasures. These notes aim to present, in a concise and self-contained form, thefundamental results on Gaussian processes and measures based on theisoperimetric tool. In particular, our exposition will include, from thismodern point of view, some of the by now classical aspects such asintegrability and tail behavior of Gaussian semi norms, large deviations orregularity of Gaussian sample paths. We will also concentrate on some of the morerecent aspects of the theory which deal with small ball probabilities. We give a martingale proof of Gaussian isoperimetry,which also contains Bobkov’s inequality on the two-point space and itsextension to non-symmetric Bernoulli measures. We derive the equivalence ofdifferent forms of Gaussian type isoperimetry. This allows us to prove a sharpform of Bobkov’s inequality for the sphere and to get new isoperimetricestimates for the unit cube.