Timeline for Isometric embeddings of metric spaces in Hilbert spaces
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
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| Jan 25, 2014 at 3:02 | review | Close votes | |||
| Jan 25, 2014 at 21:34 | |||||
| Jan 24, 2014 at 19:07 | comment | added | Alex M. | @Bill Johnson: I see that the proof found in Wells and Williams depends on Zorn's lemma. Would you happen to know whether this is just for convenience, or a fundamental aspect of this fact? (Lemma 2.3, page 3). | |
| Jan 20, 2014 at 19:00 | comment | added | Bill Johnson | @AlexM. Perhaps you should read some papers that use Shoenberg's theorem. I suggest the papers of Aharoni, Maurey and Mitiagin that use it to classify the Banach spaces that uniformly embed into a Hilbert space and the one by Randrianarivony that classifies the Banach spaces that coarsely embed into a Hilbert space. We do not have analogous classifications of Banach spaces that uniformly or coarsely embed into $L_p$ for other $p$ simply because there is no replacement for Schoenberg's theorem. | |
| Jan 20, 2014 at 18:28 | vote | accept | Alex M. | ||
| Jan 20, 2014 at 18:20 | comment | added | Alex M. | @alvarezpaiva mentions some results of Frechet and Menger. Are they along the same lines as those of Schoenberg mentioned by Bill Johnson? | |
| Jan 20, 2014 at 18:19 | comment | added | Alex M. | @Bill Johnson's bibliographic indication (Wells and Williams, "Embeddings and extensions in analysis", Springer Ergebnisse 84) answers my problem, I guess, even though the characterization of those metric spaces that can be embedded in Hilbert spaces from this book is quite technical and not very enlightening. | |
| Jan 20, 2014 at 18:18 | history | edited | Alex M. | CC BY-SA 3.0 |
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| Jan 20, 2014 at 14:44 | comment | added | alvarezpaiva | @AlexM. Benoit already answered your comment (why the induced metric and the induced length metric are not the same). I just wanted to add that while the question of when a metric space can be embedded in a Hilbert space was settled by Frechet, Schoenberg, and Menger, the existence of length preserving embeddings between length spaces and Banach space has not been studied much. Sample problem: does every compact Finsler surface admit an isometric embedding into $L_1$? | |
| Jan 20, 2014 at 10:59 | comment | added | Benoît Kloeckner | Alex M, the unit sphere in Euclidean space has two natural induced "metrics": the induced distance (aka chordal metric) and the distance induced by the induced Riemannian metric. They are quite different. For connected manifolds, one is the length metric associated to the other. You should probably clarify the question by explicitely writing what kind of embedding you are looking for (probably length-preserving). Also, your first paragraph seems to obfuscate this point a little further, as length preserving embeddings into Banach space seem unusual. | |
| Jan 20, 2014 at 6:50 | comment | added | j.c. | re: the notion of isometric embedding, see math.stackexchange.com/questions/87503/isometric-embedding | |
| Jan 19, 2014 at 23:31 | comment | added | Anton Petrunin | @GeraldEdgar, it is a legth-preserving embedding. It is a historical term, when it was invented (Gauss?) people probably thought that grobal isometric embedding are too boring. (I also hate that isometric embeddings are not isometric...) | |
| Jan 19, 2014 at 23:11 | comment | added | Gerald Edgar | Interesting. What is the definition of "isometric embedding" used in differential geometry? And how can someone reading the question realize that it does not mean a global isometric embedding? | |
| Jan 19, 2014 at 22:46 | answer | added | Alexandre Eremenko | timeline score: 4 | |
| Jan 19, 2014 at 20:34 | comment | added | Anton Petrunin | @BillJohnson I am sure you did not read the question to the end. Schoenberg's results (which are by the way totally trivial) are about GLOBAL isometric embedding and OP asks about isometric embeddings as it is usually defined in diff. geometry. | |
| Jan 19, 2014 at 20:07 | comment | added | Alex M. | @alvarezpaiva: I might be superficial here, but aren't the two concepts identical (through the Myers-Steenrod theorem)? | |
| Jan 19, 2014 at 20:06 | comment | added | Alex M. | @Bill Johnson: I'm sorry if things are as you say. I assure you that I have first searched the whole web, not just MO, for the answer, before asking my question. I have found only one slightly related question on MO, migrated on SE. | |
| Jan 19, 2014 at 20:01 | review | Close votes | |||
| Jan 20, 2014 at 6:53 | |||||
| Jan 19, 2014 at 19:44 | comment | added | Bill Johnson | This question appears to be off-topic because it is about a topic that has been well covered on MO and the OP could have found the answer easily with a Google search. | |
| Jan 19, 2014 at 19:43 | comment | added | Bill Johnson | Schoenberg's 1935 and 1938 papers address your question. If you do not want to go to the original papers, read about it e.g. Wells and Williams, Embeddings and extensions in analysis, Springer Ergebnisse 84. | |
| Jan 19, 2014 at 19:01 | answer | added | Anton Petrunin | timeline score: 7 | |
| Jan 19, 2014 at 18:54 | comment | added | alvarezpaiva | The isometric embedding theorems in Banach spaces are of a very different nature: they are distance preserving maps. Nash theorem is about length preserving maps. | |
| Jan 19, 2014 at 18:45 | review | First posts | |||
| Jan 19, 2014 at 18:48 | |||||
| Jan 19, 2014 at 18:25 | history | asked | Alex M. | CC BY-SA 3.0 |