Show the following map is jointly convex.

Question

The map $(A,B) \rightarrow Tr[A \log(A)] - Tr[A\log(B)]$ from $H_n^+ \times H_n^+$ to $R$ is jointly convex.

I have the following proof but I do not understand it. I can write it, but I would appreciate it if you could explicitly explain or give another solution to it.

For all $0<p<1 , (A,B) \rightarrow Tr(B^{1-p}A^p)$, we show $(a,b)$ is convex and that convexity is conserved.

Answer 1

It is three steps proof:

1. Apply Lieb's concavity theorem to conclude that $$ (A,B)\mapsto \text{Tr}\,(B^{1-p}A^p) $$ is jointly concave.
2. Construct a new function $$ (A,B)\mapsto \underbrace{\frac{1}{p-1}}_{\text{negative}}\Big(\underbrace{\text{Tr}\,(B^{1-p}A^p)-\text{Tr}\,A}_{\text{concave}}\Big). $$ It is convex (=minus concave).
3. Limit of convex functions is convex. Note that the limit of the function i Step 2 when $p\to 1$ is the derivative of the function at $p=1$. Take the derivative w.r.t. $p$ $$ (A,B)\mapsto \text{Tr}\,(-B^{1-p}\log(B)\, A^p+B^{1-p}A^p\log(A)) $$ and set $p=1$. Change the order of $A$ and $\log(B)$ under trace (possible). Done.