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?