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 $K_y$.


  • Did anyone consider this construction?
  • Does it have a name?
  • 2
    $\begingroup$ Isn't this just a projection of a convex set in $E\times E$ onto a certain quotient space? $\endgroup$ – Robin Chapman Jun 4 '10 at 13:21
  • $\begingroup$ Up to a factor of $\sqrt{2}$, yes. $\endgroup$ – Mark Meckes Jun 4 '10 at 13:32
  • $\begingroup$ @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... $\endgroup$ – Anton Petrunin Jun 4 '10 at 14:00
  • $\begingroup$ @Anton: What kind of more general situation? $\endgroup$ – François G. Dorais Jun 4 '10 at 16:43
  • $\begingroup$ @François, I need some kind of arithmetic in tangent cone of Alexandrov space. $\endgroup$ – 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.

  • $\begingroup$ Do you have a reference or two? I find this notation rather strange, so I'd like to see it in context. $\endgroup$ – François G. Dorais Jun 4 '10 at 16:50
  • 1
    $\begingroup$ @François: I've added a couple of links. $\endgroup$ – Kevin O'Bryant Jun 4 '10 at 21:24
  • 1
    $\begingroup$ Thank you, "sumset" sounds nice. By accident it was the first name which I came up with... $\endgroup$ – Anton Petrunin Jun 4 '10 at 22:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.