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Right, that's why I moved the part about finiteness into the formal definitions section, and left the introductory sections talking about metric spaces more generally. I am not certain whether the definition here is correct for infinite spaces, whether perhaps the condition on existence of y s.t.
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Yes, that's one of the reasons the connection to orthogonal hull is hard to generalize to higher dimensions. I suppose I should say more about how two-dimensional L1 and Linf are related somewhere in that example.
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Develin & Sturmfels (2004) provide an alternative definition of the tight span of a finite metric space, as the tropical convex hull of the vectors of distances from each point to each other point in the
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f(x)+f(y)=d(x,y) should be replaced by inf f(x)+f(y)-d(x,y)=0, or whether something else is required in the infinite case, so I thought it best to stick to material I was more certain I understood. —
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article. Rather than expunging the error, it's reasonable to show that doing mathematics (or any other science) is an ongoing process, subject to correction when we err.
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The tight span is defined for all metric spaces, not just finite metric spaces, although the applications that I have seen all seem to be for finite metric spaces.
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Holsztyński, Włodzimierz (1968). "Linearisation od isometric embeddings of Banach Spaces. Metric
Envelopes.". Bull. Acad. Polon. Sci. 16: 189-193.
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Besides the k-server problem, what other published applications to online algorithms do you know of? I didn't find any when I was looking.
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I don't know of any, but I think that line is left over from Oravec's earlier version, so you could try asking him...—
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always contains Isbell’s injective hull of the metric, but in general these two polyhedral spaces are not equal.
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Later: I found the right definition in Dress et al (it is the inf version) and added it as a footnote. —
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I see. For example, the tight span of the rationals is the reals, and you really need to say "inf."
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isometric embeddings of Banach Spaces. Metric
Envelopes.". Bull. Acad. Polon. Sci. 16: 189-193.
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to their paper seems to retract the theorem that the tropical convex hull is the tight span.
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is a symmetric matrix which represents a finite metric, then the tropical polytope
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Wow, you guys developed this article nicely in a short period of time. Congrats.
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I believe that in the T-theory literature, the tight span of
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126:{\displaystyle T_{X}}
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99:{\displaystyle T(X)}
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457:Unassessed articles
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39:Who Discovered It?
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304:References
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