  We present new stochastic geometry theorems that give bounds on the
probability that $m$ random data classes all contain a point in common in their
convex hulls. We apply these stochastic separation theorems to obtain bounds on
the probability of existence of maximum likelihood estimators in multinomial
logistic regression. We also discuss connections to condition numbers for
analysis of steepest descent algorithms in logistic regression and to the
computation of centerpoints of data clouds.
