# A variation of Minkowski sum

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?
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Isn't this just a projection of a convex set in $E\times E$ onto a certain quotient space? –  Robin Chapman Jun 4 '10 at 13:21
Up to a factor of $\sqrt{2}$, yes. –  Mark Meckes Jun 4 '10 at 13:32
@Robin, sure, but I need much more general thing, where no projections can be defined. Mostly I think what would be right way to call such thing... –  Anton Petrunin Jun 4 '10 at 14:00
@Anton: What kind of more general situation? –  François G. Dorais Jun 4 '10 at 16:43
@François, I need some kind of arithmetic in tangent cone of Alexandrov space. –  Anton Petrunin Jun 4 '10 at 21:57

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 of Alon, Angel, Benjamini, and Lubetzky. Also, a google scholar search shows the terminology in action.

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Do you have a reference or two? I find this notation rather strange, so I'd like to see it in context. –  François G. Dorais Jun 4 '10 at 16:50
@François: I've added a couple of links. –  Kevin O'Bryant Jun 4 '10 at 21:24
Thank you, "sumset" sounds nice. By accident it was the first name which I came up with... –  Anton Petrunin Jun 4 '10 at 22:03