  The entropy power inequality, which plays a fundamental role in information
theory and probability, may be seen as an analogue of the Brunn-Minkowski
inequality. Motivated by this connection to Convex Geometry, we survey various
recent developments on forward and reverse entropy power inequalities not just
for the Shannon-Boltzmann entropy but also more generally for R\'enyi entropy.
In the process, we discuss connections between the so-called functional (or
integral) and probabilistic (or entropic) analogues of some classical
inequalities in geometric functional analysis
