I'm doing the exercise below and I'm stuck on how to show what is being asked for.
Given an $m \times n$ matrix $A$ and a convex set $ C \in \mathbb{R}^n$, show that the set $\{Ax |x \in C\}$ is convex.
In this case, I also want to show or give a counter example if I replace the word convex with open, closed and compact. However using the definition of convex set $\alpha x + (1 – \alpha)y \in A$ for the first case, I tried to do the following:
- Like $x \in C$ then $\alpha Ax + (1 – \alpha)y \in C$
But I don't know if this is correct or how to continue.
Can someone help me?
Best Answer
You want to show that $$\text{for }\lambda\in[0,1],\ A\tilde x,A\tilde y\in Ax:\; \lambda A\tilde x +(1-\lambda) A\tilde y\in Ax$$So: $$\lambda A\tilde x +(1-\lambda) A\tilde y=A(\lambda \tilde x +(1-\lambda)\tilde y)$$And because $\tilde x,\tilde y\in C$ and C is convex, we know that $\lambda \tilde x +(1-\lambda)\tilde y\in C$ and so $A(\lambda \tilde x +(1-\lambda)\tilde y)\in Ax$, meaning $Ax$ is convex