  For systems of an arbitrary dimension, a theory of geometric chained Bell
inequalities is presented. The approach is based on chained inequalities
derived by Pykacz and Santos. For maximally entangled states the inequalities
lead to a complete $0=1$ contradiction with quantum predictions. Local realism
suggests that the probability for the two observes to have identical results is
$1$ (that is a perfect correlation is predicted), whereas quantum formalism
gives an opposite prediction: the local results always differ. This is so for
any dimension. We also show that with the inequalities, one can have a version
of Bell's theorem which involves only correlations arbitrarily close to perfect
ones.
