  The classical isodiametric inequality in the Euclidean space says that balls
maximize the volume among all sets with a given diameter. We consider in this
paper the case of Carnot groups. We prove that for any Carnot group equipped
with a Haar measure one can find a homogeneous distance for which this fails to
hold. We also consider Carnot-Caratheodory distances and prove that this also
fails for these distances as soon as there are length minimizing curves that
stop to be minimizing in finite time. Next we study some connections with the
comparison between Hausdorff and spherical Hausdorff measures, rectifiability
and the generalized 1/2-Besicovitch conjecture giving in particular some cases
where this conjecture fails.
