Cartesian product of convex sets is convex

The product Ω1×Ω2 is convex when each factor is convex

Cartesian product of convex sets is convex

Proposition. Let $\Omega_1\subset X$ and $\Omega_2\subset Y$ be convex sets in vector spaces $X$ and $Y$ . Then the Cartesian product $\Omega_1\times \Omega_2$ is convex in the product space $X\times Y$ .

Context. Convexity is stable under forming product constraints, a basic fact used in multi-variable convex analysis.

Proof sketch. If $(x_1,y_1),(x_2,y_2)\in\Omega_1\times\Omega_2$ and $\lambda\in[0,1]$ , then

\lambda(x_1,y_1)+(1-\lambda)(x_2,y_2)=(\lambda x_1+(1-\lambda)x_2,\ \lambda y_1+(1-\lambda)y_2),

and each component lies in the appropriate convex set.