Concavity of mutual information in the input distribution

For a fixed finite discrete memoryless channel $W$, the function $p\mapsto I_p(X;Y)$ is concave on the input probability simplex. Indeed, the induced output law $q=pW$ is linear in $p$, the output Shannon entropy $H(q)$ is concave, and $H(Y\mid X)={\sum }_{x}p(x)H(W(\cdot \mid x))$ is linear. Their difference is therefore concave. Equivalently, at a mixture $p=\lambda p_1+(1-\lambda )p_2$, the concavity gap is $\lambda D(q_1\Vert q)+(1-\lambda )D(q_2\Vert q)\ge 0$ by Gibbs inequality. Thus every local maximum is a global maximum, and averaging over symmetries of the fixed channel cannot decrease mutual information.