A simple "a posteriori" criterion is that on the optimal tour the distances to the tour-neighbors is smaller than that to any of the other vertices.  
Convexity alone doesn't suffice as the example of ellipses with sufficiently high excentricity demonstrates.  

Another, "a priory" criterion *may* be that the maximum weight matching of every $K_4$ induced by 4 vertices of the TSP instance consists of the two longest edges of that subgraph.  

Ruminating further, a general sufficient condition is that the Minimum Spanning Tree is linear and the edge joining the leaf nodes doesn't cross a tree edge.  

The simplest sufficient and generally applicable criterion is however that the set of edges that is the union of the two shortest edges, that are adjacent to a vertex, constitutes to a tour.