An isoperimetric inequality is a lower bound on the boundary measure of sets
in terms of their measure. Finding the optimal sets (of given measure and of
minimal boundary measure) is very difficult, and the only hope is to estimate
the isoperimetric function. This is well understood on the line (Bobkov) and
for the product of standard Gaussian measures (Sudakov-Tsirel'son, Borell).
We shall recall these known results.
Then, we shall explain how functional inequalities can be used to get
dimension free isoperimetric inequalities for measures between exponential
and Gaussian. Also, using the transport of mass technique we shall dervive
isoperimetric inequalities (depending on the dimension) for measures with
tails larger than exponential.