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Minimizing divergence measures under a constraint is an important problem. Wederive a sufficient condition that binary divergence measures provide lowerbounds for symmetric divergence measures under a given triangulardiscrimination or given means and variances. Assuming this sufficientcondition, the former bounds are always tight, and the latter bounds are tightwhen two probability measures have the same variance. An application of theseresults for nonequilibrium physics is provided.

On Relations Between Tight Bounds for Symmetric $f$-Divergences and Binary Divergences