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Introduced by Chatterjee and Dembo, the nonlinear large deviation theoryÌýaims at unifying under the same paradigm certain large deviations problemsÌýsuch as the problem of the upper tail of sub-graph counts in Erdös-RényiÌýgraphs, or of the traces of powers of Wigner matrices. This paradigmÌýconsists in the fact that for these large deviations problems, the optimalÌýlarge deviations strategy corresponds to changes of measure which have anÌýaffine log-density with respect to the background measure.The goal of theÌýnonlinear large deviations theory is to find a sufficient criterion for thisÌýparticular strategy to be optimal in a given large deviation problem, and toÌýpropose quantitative estimates. We will discuss some improvements on thisÌýquestion which will lead us to develop transportation tools to prove in theÌýcase of the Gaussian measure and the uniform measure on discrete hypercubeÌýdimension-free estimates.