I'm interested in


$$\sup\_{x, y} \left\| A x + B y\right\|\_2$$ subject to 
$$\left\|x\right\|\_2 = \left\|y\right\|\_2 = 1$$


where $A$, $B$ and $x$, $y$ are real matrices and vectors, respectively, of compatible sizes. (Generalizations to more than two terms are also interesting)


Does this problem have a name in the literature? (If not, does it reduce to a well-known one?)




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Note that this is a stronger constraint than $\left\|\begin{bmatrix}x\\y\end{bmatrix}\right\|\_2 = \sqrt 2$



$\sup \left\| A x + B y\right\|_2$ subject to $\left\|x\right\|_2 = \left\|y\right\|_2 = 1$
The sets $\{Ax : \|x\|=1\}$ and $\{By : \|y\|=1\}$ are ellipsoids. Hence the set $\{Ax+By : \|x\|=\|y\|=1\}$ is the [Minkowski sum](https://en.wikipedia.org/wiki/Minkowski_addition) of two ellipsoids. Googling for these terms returned this paper which may be interesting to you (although browsing through it I did not find an immediate solution to your problem). 


*Yan, Yan; Chirikjian, Gregory S.*, [**Closed-form characterization of the Minkowski sum and difference of two ellipsoids**](http://dx.doi.org/10.1007/s10711-014-9981-3), Geom. Dedicata 177, 103-128 (2015). [ZBL1321.65033](https://zbmath.org/?q=an:1321.65033).


