Convex variant of Bhattacharyya coefficient

Question

For (discrete, finite) probability distributions $P,Q$, the Bhattacharyya coefficient is $B(P,Q) := \sum_x \sqrt{P_x Q_x}$. It can be shown that this is jointly concave in $P$ and $Q$. My question is, is there a monotone increasing function of the Bhattacharyya coefficient that is instead jointly convex in $P$ and $Q$? For instance, is it possible that $B(P,Q)^k$ might be jointly convex for some sufficiently large $k$? (As a starting point, it can be shown that $k=2$ is not sufficient.)

As a follow-up, does the joint convexity of that variant also hold if its domain is extended to include subnormalized probability distributions as well?

(The context of the question is that I wish to minimize $B(p,q)$ for subnormalized distributions $p$, $q$ belonging to convex sets $\mathcal{S}$ and $\mathcal{T}$ respectively. If I can transform the objective (in a monotone increasing fashion) to a convex function, then I would be able to apply standard convex-optimization algorithms.)

Answer 1

In hindsight, this question has an obvious negative answer, apart from the trivial case where the transformed function is simply a constant function. Suppose there exists an increasing function $f: [0,1] \to \mathbb{R}$ such that the function $\tilde{B}(P,Q) := f(B(P,Q))$ is jointly convex with respect to $(P,Q)$. Pick any distribution $R$ in the interior of the set of distributions being considered, in which case the point $(P,Q)=(R,R)$ lies in the interior of the domain of $\tilde{B}$. Now note that we have $\tilde{B}(R,R) = f(B(R,R)) = f(1)$, and since $f$ is increasing on $[0,1]$, this means $\tilde{B}$ attains its maximum possible value (over its domain) at an interior point of its domain. But the only convex functions that can do so are constant functions, implying that $\tilde{B}$ can only be a constant function.