Attainability Problem for triangles (n=3)

Determine, for n = 3, the set of triangles P' contained in a given triangle P that are attainable from P via a decreasing path of polygons, meaning a continuous path of triangles whose convex hulls form a decreasing family of sets. Equivalently, characterize all triangles P' with co(P') ⊆ co(P) that can be reached from P by such a decreasing path.

Background

The paper introduces the Attainability Problem: given an n-gon P, describe all n-gons P' contained in P that can be reached from P by a decreasing path, where a decreasing path is a continuous path of polygons whose convex hulls are nonincreasing along the path. The authors fully solve this problem for n ≥ 4 using the Poncelet map, the broken line construction, and the notions of a threshold and vestibule region.

For n = 3 (triangles), the problem connects to the embeddability problem for non-homogeneous Markov processes. Prior work shows that any attainable triangle is reachable by finitely many pull-in moves, but a complete characterization of which triangles are attainable from a given triangle via decreasing paths is not provided here; the authors explicitly note this case remains open.