I have to work with the following variation of Minkowski sum:



> 
> Let $\mathbb E$ be a Euclidean space and $K$ be a convex set in $\mathbb E\times \mathbb E$.
>  Set 
>  $$K^+=\{\\,x+y\in\mathbb E\mid(x,y)\in K\\,\}.$$
> 
> 
> 


Note that if $K=K\_x\times K\_y$ for some convex sets $K\_x$ and $K\_y$ in $\mathbb E$ then $K^+$ is the usual Minkowski sum of $K\_x$ and $K\_y$.


**Questions:** 


* Did anyone consider this construction?
* Does it have a name?



A variation of Minkowski sum
In additive combinatorics, we call the Minkowski sum the sumset, and write it as ${\mathbb E}+{\mathbb E}$. We call what you're talking about the "sumset along a graph", and write it as ${\mathbb E}+\_K{\mathbb E}$, where $K$ is any graph (you call it a subset of ${\mathbb E}\times {\mathbb E}$ and I call it a graph, but it's the same thing!).


For an example of this terminology in use, check out [this paper](http://arxiv.org/abs/0905.0135) of Alon, Angel, Benjamini, and Lubetzky. Also, a [google scholar search](http://www.google.com/search?q=sumsets+along+a+graph) shows the terminology in action.


