Given two compact convex set $X \subset \mathbf{R}^2, Y \subset \mathbf{R}^2 $.
The Cartesian product $Z := X \times Y \subset \mathbf{R}^4$ Is again a convex subset.
Is the low dimension projection set $A = \{(x_1, y_1)|(x_1, x_2, y_1, y_2) \in Z\}$, or $B = \{(x_2, y_1)|(x_1, x_2, y_1, y_2) \in Z\}$, or any other projection still convex ?
Best Answer
Yes, convexity is preserved under images of affine functions.
Proof: Suppose $C \subset \mathbb{R}^m$ is convex and $f: \mathbb{R}^m \to \mathbb{R}^n$ is affine, so $f$ can be written as $f(x) = Ax+b$. Then for $y_1, y_2 \in f(C)$ and $\lambda \in [0,1]$, there exist $x_1, x_2$ such that $f(x_1) = y_1$ and $f(x_2) = y_2$.
Thus,
$\begin{align*}\lambda y_1 + (1- \lambda)y_2 &= \lambda f(x_1) + (1-\lambda)f(x_2)\\ &= f(\lambda x_1) + f((1 - \lambda) x_2) \\&= f(\lambda x_1 + (1-\lambda) x_2) \in f(C)\end{align*}$
It follows $f(C)$ is convex. $\square$
Since your projection is linear, its image of a convex set will be convex.