# Existence of $1$-Lipschitz map between triangles

Crosspost from math.SE

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).

It follows from Kirszbraun‘s theorem on the extension of Lipschitz mappings that a necessary and sufficient condition is that they can be labelled as $$ABC$$ and $$DEF$$ so that $$|DE|\leq|AB|$$ etc.
• Remark that I ask that sides are sent to sides. With this condition, your condition is not sufficient: take a triangle with side 1, 1 and 0. Then the triangle with side 1, 1 and $\epsilon$ cannot majorize it (for the obvious labeling). – Florentin MB Jun 12 '19 at 12:21