  Bi-log-concavity of probability measures is a univariate extension of the
notion of log-concavity that has been recently proposed in a statistical
literature. Among other things, it has the nice property from a modelisation
perspective to admit some multimodal distributions, while preserving some nice
features of log-concave measures. We compute the isoperimetric constant for a
bi-log-concave measure, extending a property available for log-concave
measures. This implies that bi-log-concave measures have exponentially
decreasing tails. Then we show that the convolution of a bi-log-concave measure
with a log-concave one is bi-log-concave. Consequently, infinitely
differentiable, positive densities are dense in the set of bi-log-concave
densities for $L_p-$norms, $p \in [1;+\infty]$. We also derive a necessary and
sufficient condition for the convolution of two bi-log-concave measures to be
bi-log-concave. We conclude this note by discussing ways of defining a
multi-dimensional extension of the notion of bi-log-concavity. We propose an
approach based on a variant of the isoperimetric problem, restricted to
half-spaces.
