Consider the case where the cities are given as points (x,y) in the plane and the distance function is the regular Euclidean distance, i.e.,
In NP one can guess the ordering of the cities, the problem is in checking whether the sum of distances is at most k. Since we can approximate the square root to a polynomial number of digits the issue is how many digits of accuracy we need. So it boils down to the following purely mathematical question: Given an expression of length m over integers with addition, subtraction, squares and square roots (but no recursive squares or square roots) what is the smallest positive number than can be expressed? If the answer is at least 2-mk for some k then TSP on the plane is in NP and NP-complete.
Let me also mention that one can well approximate traveling salesman on the plane.