An inequality on the edge lengths of triangular meshes

We give a short proof of the following geometric inequality: for any two triangular meshes A and B of the same polygon C, if the number of vertices in A is at most the number of vertices in B, then the maximum length of an edge in A is at least the minimum distance between two vertices in B. Here the vertices in each triangular mesh include the vertices of the polygon and possibly additional Steiner points. The polygon must not be self-intersecting but may be non-convex and may even have holes. This inequality is useful for many purposes, especially in proving performance guarantees of mesh generation algorithms. For example, a weaker corollary of the inequality confirms a conjecture of Aurenhammer et al. Theoretical Computer Science 289 (2002) 879-895] concerning triangular meshes of convex polygons, and improves the approximation ratios of their mesh generation algorithm for minimizing the maximum edge length and the maximum triangle perimeter of a triangular mesh.