  The support of minimizing measures of the causal variational principle on the
sphere is analyzed. It is proven that in the case $\tau>\sqrt{3}$, the support
of every minimizing measure is contained in a finite number of real analytic
curves which intersect at a finite number of points. In the case
$\tau>\sqrt{6}$, the support is proven to have Hausdorff dimension at most
$6/7$.
