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There is a related question here (with an answer similar to Andrew's --- it answers question #1).

Question #2a: NO, say take 4 points p,x,y,z such that |px|=|py|=|pz|=1 and $|xy|=|yz|=|zx|=2$.

Question #2b: YES, there is a trivial embedding into metric graph, which can be approximated arbitrary well by graph with fixed length of edges.

Question #0: Some related questions appear when one plays with definition of Alexandrov space, check also Gromov's CAT(κ)-Spaces: Construction and Concentration http://www.springerlink.com/content/m1275p3g0642700l/

There is a related question here (with an answer similar to Andrew's --- it answers question #1).

Question #2a: NO, say take 4 points p,x,y,z such that |px|=|py|=|pz|=1 and $|xy|=|yz|=|zx|=2$.

Question #2b: YES, there is a trivial embedding into metric graph, which can be approximated arbitrary well by graph with fixed length of edges.

Question #0: Some related questions appear when one plays with definition of Alexandrov space, check also Gromov's CAT(κ)-Spaces: Construction and Concentration Link

There is a related question here (with an answer similar to Andrew's --- it answers question #1).

Question #2a: NO, say take 4 points p,x,y,z such that |px|=|py|=|pz|=1 and $|xy|=|yz|=|zx|=2$.

Question #2b: YES, there is a trivial embedding into metric graph, which can be approximated arbitrary well by graph with fixed length of edges.

Question #0: Some related questions appear when one plays with definition of Alexandrov space, check also Gromov's CAT(κ)-Spaces: Construction and Concentration http://www.springerlink.com/content/m1275p3g0642700l/

There is a related question here (with an answer similar to Andrew's --- it answers question #1).

Question #2a: NO, say take 4 points p,x,y,z such that |px|=|py|=|pz|=1 and $|xy|=|yz|=|zx|=2$.

Question #2b: YES, there is a trivial embedding into metric graph, which can be approximated arbitrary well by graph with fixed length of edges.

Question #0: Some related questions appear when one plays with definition of Alexandrov space, check also Gromov's CAT(κ)-Spaces: Construction and Concentration http://www.springerlink.com/content/m1275p3g0642700l/

There is a related question here (with an answer similar to Andrew's --- it answers question #1).

Question #2a: NO, say take 4 points p,x,y,z such that |px|=|py|=|pz|=1 and $|xy|=|yz|=|zx|=2$.

Question #2b: YES, there is a trivial embedding into metric graph, which can be approximated arbitrary well by graph with fixed length of edges.

Question #0: There are some very related questions in Alexandrov geometry, look in papers of Berestovski

There is a related question here (with an answer similar to Andrew's --- it answers question #1).

Question #2a: NO, say take 4 points p,x,y,z such that |px|=|py|=|pz|=1 and $|xy|=|yz|=|zx|=2$.

Question #2b: YES, there is a trivial embedding into metric graph, which can be approximated arbitrary well by graph with fixed length of edges.

Question #0: Some related questions appear when one plays with definition of Alexandrov space, check also Gromov's CAT(κ)-Spaces: Construction and Concentration http://www.springerlink.com/content/m1275p3g0642700l/