  Let $G$ be a compact, simply connected Lie group. If
$\mathcal{C}_1,\mathcal{C}_2$ are two $G$-conjugacy classes, then the set of
elements in $G$ that can be written as products $g=g_1g_2$ of elements $g_i\in
\mathcal{C}_i$ is invariant under conjugation, and its image under the quotient
map $G\to G/\operatorname{Ad}(G)$ is a convex polytope inside the Weyl alcove.
In this note, we will prove an analogous statement for twisted conjugations
relative to group automorphisms. The result will be obtained as a special case
of a convexity theorem for group-valued moment maps which are equivariant with
respect to the twisted conjugation action.
