Lipschitz Gromov-Hausdorff appr oximat ions to two-dimensional closed piecewise flat Alexandrov spaces

We show that a closed piecewise fiat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an application, we show that for any metric space sufficiently close to Σ in the Gromov-Hausdorff topology, there is a Lipschitz Gromov-Hausdorff approximation.