  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 of
Bobkov-Chistyakov (2015) fails when the R\'enyi parameter $r\in(0,1)$, we show
that random vectors with $s$-concave densities do satisfy such a R\'enyi
entropy power inequality. Along the way, we establish the convergence in the
Central Limit Theorem for R\'enyi entropies of order $r\in(0,1)$ for
log-concave densities and for compactly supported, spherically symmetric and
unimodal densities, complementing a celebrated result of Barron (1986).
Additionally, we give an entropic characterization of the class of $s$-concave
densities, which extends a classical result of Cover and Zhang (1994).
