What is known about this TSP variant?
Huck Bennett has answered this question on cstheory.SE here.


  This is a special case of the Travelling Salesman with Neighborhoods
  (TSPN) problem. In the general version, the neighborhoods need not all
  be the same.
  
  A paper by Dumitrescu and Mitchell, Approximation algorithms for TSP
  with neighborhoods in the plane, addresses your question. They
  give a constant factor approximation algorithm for a slightly more
  general problem (case 1), and a PTAS when the neighborhoods are
  disjoint balls of the same size (case 2).
  
  As a side comment, I think Mitchell has done a lot of work on
  geometric TSP variants, so you might want to look at his other papers.