Proving that the ball is converx

Question

I need to prove that the ball $B(x,r)=\{y\in \mathbb{R^n}:||y-x||<r\}$ is convex. How to do this?

Answer 1

1. Prove generally that convexity is stable under affine linear transformations. In particular, show that for any $x ∈ ℝ^n$, a set $C ⊂ ℝ^n$ is convex if and only if its translation $x + C ⊂ ℝ^n$ is convex. This is generally useful.

2. By this, you have reduced to the case of proving $B(0,r)$ is convex. (Since then $B(x,r) = x + B(0,r)$ is convex, too.) Now, for $λ ∈ [0..1]$ and $v ∈ B(0,r)$ use the triangle inequality on $λv + (1-λ)v$ to determine whether it’s in $B(0,r)$ or not.

If you don’t show 1., you can still show 2. by generalizing the argument for the convexity of $B(0,r)$. However, in doing so, you’d essentially prove 1. as well.