A conditional entropy can be expressed in the following way, H_{V_t}(V_s) = -\sum_{s,t}p(s,t)\log{p_t(s)} = -\sum_{s,t}p(s,t)\log{\frac{p(s,t)}{\sum_{s'}{p(s',t)}}}
$s$ and $t$ are defined on finite domain, say $s,t\in{\{00,01,11,10\}}$. Each value of $p(s,t)$ is an independent variable. Is conditional entropy a convex function under this circumstances?