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Some context to my question: This is a part from a question about memoryless channel. The channel is given by $P_{Y\mid X}$, and $Q_Y$ is an arbitrary distribution on the channel outputs. in part of the solution for this question (the question itself is not relevant), they do the following move: $$max_{P_X}\sum_{x\in X}[P_X(x)D_{KL}(P_{Y\mid X=x}\mid\mid Q_Y)]= max_{x\in X}D_{KL}((P_{Y\mid X=x}\mid\mid Q_Y)$$

and they explain that it's true since

$max_{P_X}$ yields $P_X= 𝟙_{x=x_m}$ where $x\in X$ is maximizing the KL divergence

I don't really understand why it true, and I didn't understand the explanation.

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