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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).

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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.

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  • $\begingroup$ 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). $\endgroup$ – Florentin MB Jun 12 '19 at 12:21
  • $\begingroup$ @FlorentinMB Indeed, but at least it tells you that it is enough to map the sides and the rest will take care of itself. $\endgroup$ – fedja Jun 12 '19 at 18:03

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