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Given two discrete distributions $p, q$ which lie in an $m$-dimensional simplex, is it possible to provide a concave lower bound on the inner product between these distributions. That is we wish to find a function $f(p,q)$ such that

1) $f(p,q)$ is concave, and

2) $f(p,q) \leq \sum_{i=1}^m p_i q_i$.

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