I am sure it done somewhere, but I do not know a ref. I did something like this in my "Metric minimizing surfaces", but do not want to claim originality.

You may fix a finite set of points draw all the geodesics between them. Together these geodesics form a finite graph; they cut finitenumber of discs from your surface. Exchange each disc by the convex plane polygon provided by Reshetnyak's majorization theorem and you get an approximation.

Note that the angles of each polygon can not be smaller that the corresponding angle in the surface. Therefore the angle around each point in the approximating surface is at least $2{\cdot}\pi$; it follows that, the approximating surface in CAT[0].

By Reshetnyak's theorem, the constructed polyhedral surface admits a short map to the original one which is isometric on the finite set you started with. From this convergence follows, assuming you choose right notion of convergence for the surfaces.

I am sure it done somewhere, but I do not know a ref. I did something like this in my "Metric minimizing surfaces", but do not want to claim originality.

You may fix a finite set of points draw all the geodesics between them. Together these geodesics form a finite graph; they cut finitenumber of discs from your surface. Exchange each disc by the convex plane polygon provided by Reshetnyak's majorization theorem and you get an approximation.

Note that the angles of each polygon can not be smaller that the corresponding angle in the surface. Therefore the angle around each point in the approximating surface is at least $2{\cdot}\pi$; it follows that, the approximating surface in CAT[0].

By Reshetnyak's theorem, the constructed polyhedral surface admits a short map to the original one which is isometric on the finite set you started with. From this the convergence follows, assuming you choose right notion of convergence for the surfaces.

I am sure it done somewhere, but I do not know a ref. I did something like this in my "Metric minimizing surfaces", but do not want claim originality.

You may fix a finite set of points draw all the geodesics between them. Together these geodesics form a finite graph; they cut finitenumber of discs from your surface. Exchange each disc by plane polygon provided by Reshetnyak's majorization theorem and you get an approximation.

By Reshetnyak's theorem, the constructed polyhedral surface admits a short map to the original one which is isometric on the finite set you started with. From this convergence follows, assuming you choose right notion of convergence for the surfaces.

I am sure it done somewhere, but I do not know a ref. I did something like this in my "Metric minimizing surfaces", but do not want to claim originality.

You may fix a finite set of points draw all the geodesics between them. Together these geodesics form a finite graph; they cut finitenumber of discs from your surface. Exchange each disc by the convex plane polygon provided by Reshetnyak's majorization theorem and you get an approximation.

Note that the angles of each polygon can not be smaller that the corresponding angle in the surface. Therefore the angle around each point in the approximating surface is at least $2{\cdot}\pi$; it follows that, the approximating surface in CAT[0].

By Reshetnyak's theorem, the constructed polyhedral surface admits a short map to the original one which is isometric on the finite set you started with. From this convergence follows, assuming you choose right notion of convergence for the surfaces.