I was studying some notes on convex optimization and came across this formula
Euclidean ball with center $x_c$ and radius r:
$${\{x_c + ru ∣ ∥u∥_2 \leq 1}\}$$
How do i even go about proving that this is a convex set. I am extremely new to this and i am not asking for answers. But i just need to even know how to start proving.
I am not even sure what does this mean
On
Using the definition of convex, it is straightforward to show that if $C$ is convex then the translate $\{ x+c | c \in C\}$ is convex.
Using the definition of convex, it is straightforward to show that if $C$ is convex then the scaled set $\{ rc | c \in C\}$ is convex.
Since the norm $\| \cdot \|$ is a convex function, it is straightforward to show, from the definition, that the set $\{x | \|x\| \le 1\}$ is convex.
Verify the definition (see page 1 of the linked pdf): let $x_1, x_2\in B:={\{x_c + ru ∣ \|u\|_2 \leq 1}\}$ that is, $$x_1=x_c+ru_1\quad \text{and}\quad x_2=x_c+ru_2$$ with $\|u_1\|_2 \leq 1$ and $\|u_2\|_2 \leq 1$, and let $\theta\in[0,1]$, then show that $$\theta x_1+(1-\theta)x_2=\theta(x_c+ru_1)+(1-\theta)(x_c+ru_2)\\=x_c+r(\theta u_1+(1-\theta)u_2)$$ belongs to $B$, that is $\|\theta u_1+(1-\theta)u_2\|_2\leq 1$.