# Does maximal Bhattacharyya coefficient imply mimimal total variation distance?

I'm working on numerical optimization (linear programming), on probability distributions denoted P, Q. We want to find the minimal total variation distance and maximal Bhattacharyya coefficient between P and Q. We can quite easily find the minimal variation distance between P, Q already (via linear programming) but since the Bhattacharyya coefficient is a concave function, finding the optimal value there is harder.

We wonder, is it true that the P, Q pair that maximizes the Bhattacharyya coefficient also minimizes the total variation distance?

If this is true, this would mean that optimizing over the P, Q pairs that minimize the total variation distance should yield the P, Q pair that maximizes the Bhattacharyya coefficient. So we would have to optimize over a much smaller set making the optimization much easier.

I would really appreciate input / references.