This question continues from: Cutting convex regions into equal diameter and equal least width pieces
Definitions: The diameter of a convex region is the greatest distance between any pair of points in the region. The least width of a 2D convex region can be defined as the least distance between any pair of parallel lines that touch the region. In above post, it was also noted that for any n, any convex planar region allows partition into n convex pieces all of same diameter (these pieces could have different numbers of edges).
Question: How can any given convex n-vertex polygon be partitioned into the least number of triangular regions all with the same diameter?
Note 1: Any convex quadrilateral can be cut into 2 triangles of same diameter (if the diameter of the quad is one of its diagonals) or 3 triangles of same diameter (if the diameter of the quad is one of its sides). What we ask above is a lower bound on the number on the number of equal diameter triangles into which any given convex n-gon can be cut as a function of n (there is also the possibility that some convex n-gons might not allow partition into any number of equal diameter triangles).
Note 2: For non-convex polygons - even quadrilaterals - it is easy to see that there is no lower bound on the number of equal perimeter triangles into which the polygon can be cut.
Note 3: A similar question can be asked about cutting a convex n-gon into triangles all of same least width - and higher dimensional analogs are conceivable for these questions.
Note 4: Questions of cutting a given convex n-gon into equal area or equal perimeter triangles are more natural. The equal area case is introduced at https://en.wikipedia.org/wiki/Equidissection and the equal perimeter case is discussed in the specific case of the n-gon being a square here: https://math.stackexchange.com/questions/2822589/dissect-square-into-triangles-of-same-perimeter. We note that a square can be cut into any number of triangles all of same diameter.
