Timeline for Can a metric space with the following properties exist? Any examples?
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| 9 hours ago | comment | added | Anixx | @DenisT one can think about this aspace as of Euclidean space with "secret geodesics" or "wormholes" between points, which are shorter than conventional geodesics and of countable number for each two points. | |
| 9 hours ago | comment | added | Anixx | @DenisT I looked here en.wikipedia.org/wiki/Busemann_G-space and it seems, the axioms are compatible with the conditions in the question, if under distance we understand the supremum of the length of the geodesic paths. | |
| 9 hours ago | comment | added | Denis T | @Anixx The one which is present in absence of metric. Are you talking about Busemann spaces, G-spaces, something else? | |
| 9 hours ago | comment | added | Anixx | @DenisT I've removed "metric" but I am not sure what re-definition a geodesic needs... | |
| 9 hours ago | history | edited | Anixx | CC BY-SA 4.0 |
deleted 7 characters in body
|
| 9 hours ago | comment | added | Denis T | @Anixx So why don't you define precisely what you mean by words "metric space" and "geodesic" if you want to use them in non-classic meaning? | |
| 10 hours ago | comment | added | Anixx | @FedorPetrov classically, distance is defined as the shortest geodesic path. We can define distance as the longest geodesic path. For instance, on a sphere, not shortest but the longest arc connecting two points. Anyway, this question does not mention distance. | |
| 10 hours ago | comment | added | Fedor Petrov | @Anixx is not the length of every path (including geodesic) between two points in a metric path always not less than the distance between them? | |
| 10 hours ago | comment | added | Anixx | @DenisT Of course, the classic definition of distance between two points would be zero, but we are not talking about distance but about geodesic paths. We can define distance as the biggest geodesic path (rather than smallest as in classic metric spaces). | |
| 10 hours ago | answer | added | Tyrannosaurus | timeline score: 0 | |
| 10 hours ago | comment | added | Denis T | Maybe I'm missing some subtle detail, but it seems like property 2 forces the space in question to be a point (regardless of definition of geodesics in general metric space; I suppose they are just locally isometric maps from an interval). | |
| 10 hours ago | comment | added | Anixx | @M.Winter if it deminishes towards zero, then the condition 2 is automatically satisfied. | |
| 10 hours ago | comment | added | Anixx | @M.Winter mmm. Maybe in the space I am searching for length of the path is calculated differently, so the paths produced by your formula are deminishing as n increases. | |
| 10 hours ago | comment | added | M. Winter | Thinking a bit more, my second comment is not completely right but further conditions on $\gamma$ and $\rho$ are necessary to ensure that concatenations are geodesic. But it would perhaps be helpful to know a space that satisfies 1 + 3 (and 1 + 2 for that matter). | |
| 11 hours ago | comment | added | M. Winter | But any space satisfying 1 will have two such paths $\gamma$ and $\rho$, so violating 3, or am I getting this wrong? | |
| 11 hours ago | comment | added | Anixx | @M.Winter yes, it violates condition 3, so torus does not satisfy all these conditions. You asked for an example, satisfying only the first property. | |
| 11 hours ago | comment | added | M. Winter | I am not experienced in thinking about geodesic paths, but if between $x$ and $y$ there are two geodesic paths $\gamma$ and $\rho$, then aren't $\gamma(\rho^{-1}\gamma)^n$ also geodesic paths of arbitrarily large length, violating 3? | |
| 11 hours ago | comment | added | Anixx | @M.Winter Cliffird torus. Any torus, in fact. | |
| 11 hours ago | history | edited | M. Winter | CC BY-SA 4.0 |
added 10 characters in body; edited title
|
| 11 hours ago | comment | added | M. Winter | Do you have an example of a space with only the first property? | |
| 11 hours ago | history | asked | Anixx | CC BY-SA 4.0 |