Consider two (Euclidean) triangles $T$ and $T'$. Let's say that $T$ majorizes $T'$ if there exists a 1-Lipschitz map that sends vertices to vertices and sides to sides (for some labeling of the vertices).
My question is, what are necessary and sufficient conditions for $T$ to majorize $T'$ ?
I know a sufficient condition. Let's say that the lengths $(l_1, l_2, l_3)$ of $T$ and $(l_1', l_2', l_3')$ of $T'$ satisfy the strong triangle inequalities if $l_i + l_j - l_k \ge l_i' + l_j' - l_k'$ for all pairwise distinct $i,j,k$. Then if $T$ and $T'$ satisfy the strong triangle inequalities, then $T$ majorizes $T'$. But this condition is not necessary (see the answer on math.se).