Timeline for When does a metric space have "infinite metric dimension"? (Definition of metric dimension)
Current License: CC BY-SA 4.0
16 events
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| Jul 28, 2021 at 8:15 | history | edited | Wlod AA | CC BY-SA 4.0 |
typos, English, just minor details.
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| Aug 5, 2017 at 16:38 | comment | added | Aaron Dall | Similary, in Example 2, one obtains a rank one infinite matroid with one basis and uncountably many loops. So at least from the matroid point of view, everything is in order in these examples. | |
| Aug 5, 2017 at 13:54 | comment | added | Aaron Dall | In Example 1, the statement "every $1$-element subset of $X$ should be a basis" is only true in the spirit of loopless matroids. It's true that in this example the only metric basis is $\{0\}$, but the point is there is a (unique) rank one matroid on four elements with one basis. It is the uniform matroid $U_{1,1}$ together with three loops. | |
| Aug 4, 2017 at 14:44 | comment | added | Chill2Macht | @BenoîtKloeckner I agree (at least there is confusion in the original question, the answer may just be reflecting the question's confusion, rather than partaking in it, per se). @ WlodAA In the first example, did you mean to say "Every 2-element set $X \subset \{1, 2, 3 \}$"? I don't understand the argument if $X \subset \{ 0 , 1, 2\}$ since $\{0\}$ is already a metric generating set. | |
| Aug 3, 2017 at 20:04 | comment | added | Benoît Kloeckner | It seems there is some confusion between minimality with respect to inclusion, and with respect to cardinality. | |
| Aug 3, 2017 at 15:06 | comment | added | Chill2Macht | I agree with you about it being time to concentrate on better definitions. I asked this question as a result of your remarks: mathoverflow.net/questions/277833/… The only example I can think of for sure where metric dimension is well-defined is Euclidean space, the reasoning seems to have something to do with affine independence (i.e. any affinely independent set of $n+1$ points should be a "metric basis" for Euclidean space) -- I haven't yet put in the time to make these connections more precise. | |
| Aug 3, 2017 at 5:31 | comment | added | Wlod AA | I have added "ALSO" x 2 (one per Example). It's not hard to add more sophisticated examples. However, I feel that it's time to concentrate on better definitions, I guess. | |
| Aug 3, 2017 at 5:28 | history | edited | Wlod AA | CC BY-SA 3.0 |
Extras about the examples
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| S Aug 3, 2017 at 0:56 | history | suggested | Chill2Macht | CC BY-SA 3.0 |
added some missing words for grammar reasons
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| Aug 2, 2017 at 23:42 | review | Suggested edits | |||
| S Aug 3, 2017 at 0:56 | |||||
| Aug 2, 2017 at 23:39 | comment | added | Chill2Macht | I'm sorry about the late reply -- I didn't even notice that you had edited your answer to add more content -- I really like how you bring in matroid theory to this (since I had been working with/thinking about matroids for a completely unrelated reason). Anyway, do I understand your examples correctly? Is there a 3-element "basis" for the first example ($\{1, 2, 3 \}$) and a 2-element "basis" for the second example ($\{ x_1, x_2 \}$ for any $x_1, x_2 > 1$)? Thus "metric dimension" isn't well-defined since not all metric generating sets of minimal cardinality have the same cardinality? | |
| Aug 2, 2017 at 23:32 | vote | accept | Chill2Macht | ||
| Jul 22, 2017 at 18:23 | comment | added | Chill2Macht | Also, for a vector space, one has to show that all basises have the same number of elements, before the notion of dimension is well-defined. But I don't actually know whether each (minimal-cardinality) metric basis is actually guaranteed to have the same number of elements (or more generally cardinality), and if it isn't, then metric dimension wouldn't be well-defined. But I don't know how to verify it either way. Is this what you were referring to when you mentioned logical problems in the definitions? | |
| Jul 20, 2017 at 2:26 | history | edited | Wlod AA | CC BY-SA 3.0 |
a typo
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| Jul 19, 2017 at 5:25 | comment | added | Chill2Macht | I definitely agree with what you've written so far. I took the definition from Professor Murphy's paper, assuming that they might have some justification for the choice of terminology that was omitted from the paper. (But maybe there isn't? I don't know.) | |
| Jul 18, 2017 at 22:36 | history | answered | Wlod AA | CC BY-SA 3.0 |