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In this paper we prove a convexity property of the relative entropy alongentropic interpolations (solutions of the Schr\"odinger problem), and aregularity property of the entropic cost along the heat flow. Then we derive adimensional EVI inequality and a contraction property for the entropic costalong the heat flow. As a consequence, we recover the equivalent results in theWasserstein space, proved by Erbar, Kuwada and Sturm.

Convexity and regularity properties for entropic interpolations