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Mixtures are convex combinations of laws. Despite this simple definition, amixture can be far more subtle than its mixed components. For instance, mixingGaussian laws may produce a potential with multiple deep wells. We study in thepresent work fine properties of mixtures with respect to concentration ofmeasure and Sobolev type functional inequalities. We provide sharp Laplacebounds for Lipschitz functions in the case of generic mixtures, involving atransportation cost diameter of the mixed family. Additionally, our analysis ofSobolev type inequalities for two-component mixtures reveals natural relationswith some kind of band isoperimetry and support constrained interpolation viamass transportation. We show that the Poincar\'e constant of a two-componentmixture may remain bounded as the mixture proportion goes to 0 or 1 while thelogarithmic Sobolev constant may surprisingly blow up. This counter-intuitiveresult is not reducible to support disconnections, and appears as areminiscence of the variance-entropy comparison on the two-point space. As faras mixtures are concerned, the logarithmic Sobolev inequality is less stablethan the Poincar\'e inequality and the sub-Gaussian concentration for Lipschitzfunctions. We illustrate our results on a gallery of concrete two-componentmixtures. This work leads to many open questions.

On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities