Determining whether an edge on a mesh is convex or concave

If n_a and n_b are the normals of the both adjacent faces, and p_a and p_b vertices of the both faces that are not connected to the edge in question, wherein n_a and p_a belongs to the face A, and n_b and p_b to the face B, then

( p_b - p_a ) . n_a <= 0 => ridge edge

as well as

( p_b - p_a ) . n_a > 0 => valley edge

where "ridge edge" denotes what you have named convex edge, and "valley edge" denotes what you've named concave edge.

You can avoid numerical inaccuracies in nearly flat connections to be classified if you compute the dot-product of the both normals and compare that with a defined threshold, and do the above decision only of the threshold is exceeded.

check if opposite vertex on one of the triangle is back-facing the other triangle's plane.

e.g. triangle A, B, C, and C, B, D are linked by edge BC.

check if A is back facing or front facing the plane of [C, B, D].

Thanks, gonna try it out when I get the time :)