Extremal characteristics of tests for multiple hypotheses with given mutual total variation distances

We consider extremal values of the characteristics of nonrandomized statistical criteria in the following case: there are $n$ hypotheses with given total variation distances. It is shown that finding the extremal values of a function of error probabilities may be reduced to solving a finite-dimensional linear programming problem. We found exact formulas for extreme values of the sum of all error probabilities in the case $n=3$.