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.

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.

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

A simple "a posteriori" criterion is that on the optimal 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.

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.