Hi everybody,
We know the definition of a cone in a Real Banach Space. I want to know if there is any definition for a cone in an abstract metric space. Have you ever seen such definition anywhere?
Thanks in advance.
Hi everybody, We know the definition of a cone in a Real Banach Space. I want to know if there is any definition for a cone in an abstract metric space. Have you ever seen such definition anywhere? Thanks in advance. 


There is a natural definition of cone in the context of pointed metric spaces:
Here, $(\lambda X,p)$ denotes the metric space obtained from $(X,p)$ by multiplying the distance function by $\lambda$. Also isometric should be understood in the pointed category, i.e., $(X,x_0)$ and $(Y,y_0)$ are isometric if there exists a distance preserving map $f\colon X\to Y$ such that $f(x_0)=y_0$. This is, for example, the definition you can find in the book by Burago, Burago & Ivanov, Def 8.2.1. It also coincides with the definition in a Banach space setting. Obviously, you can then say that a subspace $(X,p)$ of your metric space $(\bar X,p)$ is a cone if it is a cone as an abstract metric space... 


First of all, the notion of $cone$ is a purely algebraic stuff, and not a metrical one. The $cone$ is naturally defined in the framework of $linear$ spaces, and not of Banach spaces. One can introduce, e.g., various "natural'' cones in a Hilbert space, without using its norm. However, if $\left(X,\, d\right)$is a complete metric space, and $\psi:X\rightarrow\left[\,0,\,\infty\right)$is a lower semicontinuous function, then the partial ordering on $X$ defined by $x\preccurlyeq y$ iff $d\left(x,y\right)\leq\psi\left(y\right)\psi\left(x\right)$is very useful in proving the CaristiKirk Fixed Point Theorem. Another metrical variant would be to use the Ralph DeMarr' cone http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.pjm/1103034358&view=body&contenttype=pdf_1, combined with an ArensEells embedding. 


Maybe this works: given a point $p$ and a subset $A$ of your metric space $(X,d)$ define the cone on $A$ from $p$ to be all points that lie between $p$ and $A$, that is all points $x$ with $d(p,a)=d(p,x)+d(x,a)$ for some $a\in A$. 


You can isometrically embed any metric space into a Banach space via the ArensEells theorem (original: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1103043959 simpler proof by E. Michael: http://www.jstor.org/stable/2034516?origin=JSTORpdf ). This embedding is, in some sense, canonical. Convex conse are welldefined in Banach spaces, so you could say that a point x is in the convex cone generated by x_1, ... x_n in the original metric space if f(x) is in the cone of f(x_1), ..., f(x_n)  where f is the ArensEells embedding. 


EDIT: As pointed out by Pete below, it seems I misunderstood the question, so what I write below is not relevant. Apologies! This is not the general answer, but in riemannian geometry there is a notion of cone. If $(M,g)$ is a riemannian manifold, then its metric cone is $\mathbb{R}^+ \times M$, with $\mathbb{R}^+$ the positive real halfline parametrised by $r>0$, with metric $$dr^2 + r^2 g$$ The best example is of course $(M,g)$ the unit sphere in $\mathbb{R}^n$ and its cone is then $\mathbb{R}^n \setminus \lbrace 0\rbrace$. In this case (and in this case only) the metric extends smoothly to the origin, but in general the apex of the cone is singular. This is used as a local model for conical (!) singularities and there is a nice interplay between the geometry of $M$ and that of its cone. The most dramatic use of the cone I know is that it turns the problem of determining which riemannian spin manifolds admit real Killing spinors into a holonomy problem, namely the determination of which metric cones admit parallel spinors. Some of this generalises to the pseudoriemannian setting; although this is perhaps not as useful as in the riaemannian setting as the holonomy classification in indefinite signatures (except for lorentzian) is still lacking. 


link textcone is a special type of set which shoud stisfy some properties. let P is cone then it shoud be closed non empty having atleast 2 elements all linera combination by positive constant shoud also lies in P if x lies in p then its assitive inverse may not lies in p 

