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We investigate the role of convexity in R\'enyi entropy power inequalities.After proving that a general R\'enyi entropy power inequality in the style ofBobkov-Chistyakov (2015) fails when the R\'enyi parameter $r\in(0,1)$, we showthat random vectors with $s$-concave densities do satisfy such a R\'enyientropy power inequality. Along the way, we establish the convergence in theCentral Limit Theorem for R\'enyi entropies of order $r\in(0,1)$ forlog-concave densities and for compactly supported, spherically symmetric andunimodal densities, complementing a celebrated result of Barron (1986).Additionally, we give an entropic characterization of the class of $s$-concavedensities, which extends a classical result of Cover and Zhang (1994).

Further investigations of R\'enyi entropy power inequalities and an entropic characterization of s-concave densities