Questions: 


* Is there some understanding of the reason, why the euclidean TSP allows a PTAS, whereas the metric TSP in general does not and, is the PTAS stable under sufficiently small perturbation of the edge weights of an euclidean TSP?
* If sufficiently small perturbations of the edge weights do not harm a PTAS, the PTAS can't be tied to euclidicity of the instance and there should be more general conditions for metric TSPs that allow a PTAS; are such conditions known or being investigated?



What is the Essential Reason that allows a PTAS for the EUCLIDEAN TSP?
One of the key ingredients in Arora's proof is the Patching Lemma, which allows you to reduce the number of crossings between subregions while expanding the length of a tour by a controlled amount. An analogue of the Patching Lemma can be proved for more general normed spaces, but not for general metric spaces.


Note, however, that although the Patching Lemma is needed if you want to apply Arora's method "out of the box", Arora himself suggests that the Patching Lemma may not be any kind of intrinsic barrier. Ultimately, there may be no more satisfying "reason" why the general metric TSP does not admit a PTAS beyond the PCP theorem.


