<s>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. </s> 

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.  

<s>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. </s>  

**Edit**  
there are counter examples to the criteria tha I have deleted: the instance with points $(0,0),\,(0,1),\,(1,1),\,(1,0),(1/k,1),\,\cdots,\,((k-1)/k,1)$