Proof:
If $S_{1}$ and $S_{2}$ are convex sets in $\mathbb R^{m\times n}$, then their partial sum:
$$S=\{(x,y_{1}+y_{2})|x\in \mathbb R^{m},y_{1},y_{2}\in\mathbb R^{n},(x,y_{1})\in S_{1},(x,y_{2})\in S_{2})\}$$
is still convex.
It seems that the convex preserving operation is needed.But I can't get the $x$ in $(x,y_{1}+y_{2})$ through $(x,y_{1})$ and $(x,y_{2})$
with the convex preserving operation.
I hope you can help me.Many thanks!
$$\ (x,y_{1}+y_{2}) = \begin{bmatrix} x \\ y_1+y_2 \\\end{bmatrix} = \begin{bmatrix} 0.5x + 0.5x \\ y_1+y_2 \\ \end{bmatrix} = \begin{bmatrix} 0.5x\\ y_1 \\ \end{bmatrix} + \begin{bmatrix} 0.5x\\ y_2 \\ \end{bmatrix} $$ Note $\begin{bmatrix} 0.5x\\ y_1 \\ \end{bmatrix} \in U_1$ and $\begin{bmatrix} 0.5x\\ y_2 \\ \end{bmatrix} \in U_2$. Here $U_1, U_2$ are convex sets. Why? Hint: Affine Transform of convex set. Do you now see convex preserving operation?