# Do averaged binary symmetric channels maximize mutual information?

This is a refined version of Do binary symmetric channels maximize mutual information?, which was answered negatively.

Let the random variables $(X, Y)$ be a doubly symmetric binary source with parameter $0 \le p \le 1/2$, i.e., $X,Y \sim \text{Bernoulli}(1/2)$ and $P(X \neq Y) = p$.

Define the two regions $\mathcal{A}, \mathcal{B} \subseteq \mathbb{R}^3$:

• $\mathcal{A}$ consists of all points $(R_0, R_1, R_2)$ such that there exist binary random variables $U,V$, satisfying the Markov chain $U - X - Y - V$ and \begin{align} R_1 &\ge \mathrm{I}(U;X) , \\ R_2 &\ge \mathrm{I}(Y;V) , \\ R_0 &\le \mathrm{I}(U;V) , \end{align} where $\mathrm{I}(\cdot;\cdot)$ is mutual information.

• $\mathcal{B}$ consists of all points $(R_0, R_1, R_2)$ such that there exist probabilities $a,b \in [0,1]$ with \begin{align} R_1 &\ge 1 - \mathrm{H}(a * p) , \\ R_2 &\ge 1 - \mathrm{H}(b * p) , \\ R_0 &\le 1 - \mathrm{H}(a * p *b) , \end{align} where $a*b := a(1-b)+(1-a)b$ is binary convolution and $\mathrm{H}()$ is the binary entropy function.

My Question: Is $\mathrm{conv}(\mathcal A) = \mathrm{conv}(\mathcal B)$, where $\mathrm{conv}()$ denotes the convex hull?

The definitions of $\mathcal{A}$ and $\mathcal{B}$ are almost the same, except that the channels $X \to U$ and $Y \to V$ are required to be symmetric for $\mathcal{B}$ and hence $\mathcal B \subseteq \mathcal A$.
The question Do binary symmetric channels maximize mutual information? asked whether $\mathcal A = \mathcal B$ and a counterexample was provided for $p=0$, which does not seem to apply here.