An alternative way of writing the rate-distortion function is: $$D(R) =\min_{p(\hat{x}|x):\ I(x;\hat{x})\leq R} \mathbb{E}[d(x,\hat{x})],$$ where $x,\hat{x}$ are variables, $d(x,\hat{x})$ is a distortion (e.g., hamming, or squared difference).
I want to prove that this is convex in $R$, which is what I saw on a source without a proof.
I can see that the function is decreasing since an increased $R$ will increase the feasible space, so decrease the minimum value. But I can't prove the convexity.