Smoothing 3-dimensional polyhedral spaces
  We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds. 
Nina Lebedeva, Vladimir Matveev, Anton Petrunin and Vsevolod   Shevchishin
math.DG math.MG