Convexity with conditional probabilities

Question

Consider the random variables $X$, $Y$, and $Z$ supported on $\mathbb{R}^3$ and with joint distribution $P_{XYZ}$.

Can I say anything about $P_{XY|Z}(x,y|z)\log P_{XY|Z}(x,y|z)$ being convex or concave on $z$?

Answer 1

In general, no, not without a lot of additional information about the functional form of $P_{XY|Z}(x,y|z)$.