I must solve this problem from Rudin's Functional Analysis book. Here $C(\mathbb{R})$ is the space of continuous complex-valued functions with real domain.
I've tried giving many examples of functions $f$ and $g$ such that both are in the ball of radius $r$ (for a given $r<1$), but I haven't been able to find a convex combination that will work. The book mentions before that such balls are not convex, so that's why I'm trying to find an example for each $r<1$. I'd appreciate any help
I also tried proving this by contradiction, assuming that the ball $B$ of radius $r^*$ was convex for a certain $0 < r^* < 1$, and I wanted to show that there was a scalar $\alpha$ such that $\alpha B$ equaled the ball of radius $1/2$ but I couldn't prove that.
