Suppose $X$ and $Y$ are $d$-dimensional set, subsets of $\mathbf{R}^N (N>>d)$
(More precisely, my case is : $X =Y= \{vec(xy^T)|x \in R^{d_1}, y \in R^{d_2}, \|x\|\leq 1, \|y\|\leq 1\}, N=d_1 d_2 $)
Can we say $Z=\{x+y| x \in X, y \in Y\}$ is a set of dimension less than $2d$? Or $Cd$ for some fixed constant C?
Can we generalize that argument for $X_1 + X_2 + \cdots + X_r$? (Something like...... it has dimension less than $rd$)
Please help!