I'd like to formulate an abstract definition of convex sets: a set $K$ is *convex* if it is endowed with a ternary operation $K\times[0,1]\times K\to K$, written $(x:t:y)$, satisfying axioms


* $(x:0:y)=(x:t:x)=x$
* $(x:t:y)=(y:1-t:x)$
* $(x:t:(y:\frac ut:z))=((x:\frac{t-u}{1-u}:y):u:z)$


The axioms imply that, for every $x\_1,\dots,x\_n\in K$ and $t\_i\ge0$ with $\sum t\_i=1$ the convex combination $\sum t\_i x\_i\in K$ is well defined.


Examples are the usual convex subsets of real vector spaces, with $(x:t:y)=(1-t)x+ty$, but also trees with $(x:t:y)$ the unique point at ratio $t$ on the geodesic from $x$ to $y$, and more generally $\mathbb R$-trees (geodesic metric spaces in which all triangles are isometric to tripods).


I'm sure this has already been explored, and I'd rather not reinvent the wheel (and I possibly missed some useful axioms), but I couldn't find any reference to such notions.


Natural questions that come to mind are:


* define an affine map between convex sets $K,L$ as a map $f\colon K\to L$ with $f(x:t:y)=(f(x):t:f(y))$. Topologize then convex sets by making all affine maps to $\mathbb R$ continuous. What can be said of these topological spaces?
* can every convex set be represented in a vector space? I have in mind the map $K\to\ell^1(K)/\{\delta\_{(x:t:y)}=(1-t)\delta\_x+t\delta\_y\forall x,t,y\}$, though the topologies will probably not match (and I'm not sure I want to close the space I quotient $\ell^1$ by).


Thanks to all! Any kinds of references or comments are welcome.



Abstract definition of convex set
There has been a bunch of work along these lines, and I think the idea has been rediscovered several times. I suggest looking at the papers of [Anna Romanowska](http://www.mini.pw.edu.pl/~aroman/www/?Publikacje), who refers to them as "barycentric algebras", to get an idea of what's known. Her book with Smith, "Modes", covers this as well as generalizations where $t$ is not required to be in $[0,1]$. [Here](http://spot.colorado.edu/~kearnes/Conf/AMSTalks/Romanowska.pdf) are some slides that cover the basics.


It's known that not all of them are representable as vector spaces. For example, if you mush everything in the interior of $[0,1]$ to a single point, your operations are still well-defined, but you can't embed it in a vector space. The Modes book has a structure theorem.


