  Elaborating on the similarity between the entropy power inequality and the
Brunn-Minkowski inequality, Costa and Cover conjectured in {\it On the
similarity of the entropy power inequality and the Brunn-Minkowski inequality}
(IEEE Trans. Inform. Theory 30 (1984), no. 6, 837-839) the
$\frac{1}{n}$-concavity of the outer parallel volume of measurable sets as an
analogue of the concavity of entropy power. We investigate this conjecture and
study its relationship with geometric inequalities.
