Sums and scalar multiples of convex sets are convex

Proposition. Let $\Omega_1,\Omega_2$ be convex subsets of a real vector space $X$ and let $\lambda\in\mathbb{R}$. Define the Minkowski sum and scalar multiple

Then $\Omega_1+\Omega_2$ and $\lambda\Omega_1$ are convex.

Context. This is a key stability property: adding convex “uncertainty sets” or scaling constraints preserves convexity.

Proof sketch. For sums, take $u_i=x_i+y_i\in\Omega_1+\Omega_2$ and use convexity of each set to show $\lambda u_1+(1-\lambda)u_2$ decomposes as a sum of two points in $\Omega_1$ and $\Omega_2$. For scalar multiples, note $\lambda(\theta x+(1-\theta)y)=\theta(\lambda x)+(1-\theta)(\lambda y)$.