  A strong converse shows that no procedure can beat the asymptotic (as
blocklength $n\to\infty$) fundamental limit of a given information-theoretic
problem for any fixed error probability. A second-order converse strengthens
this conclusion by showing that the asymptotic fundamental limit cannot be
exceeded by more than $O(\tfrac{1}{\sqrt{n}})$. While strong converses are
achieved in a broad range of information-theoretic problems by virtue of the
"blowing-up method"---a powerful methodology due to Ahlswede, G\'acs and
K\"orner (1976) based on concentration of measure---this method is
fundamentally unable to attain second-order converses and is restricted to
finite-alphabet settings. Capitalizing on reverse hypercontractivity of Markov
semigroups and functional inequalities, this paper develops the "smoothing-out"
method, an alternative to the blowing-up approach that does not rely on finite
alphabets and that leads to second-order converses in a variety of
information-theoretic problems that were out of reach of previous methods.
