Star-Convex Set Centers Form Convex Set

Question

I have the following proof that all centers of a star-convex set $U\subset\mathbb{R}^n$, $Z(U)$ form a convex set:

Suppose $Z(U)$ is not convex, then there are elements $a,b\in Z(U)$ such that the connection $[a,b] \notin Z(U)$. However by definition, $a$ is a center of $Z(U)\subset U$, therefore the connection $[a,x]$ lies in $Z(U)$ $\forall x\in Z(U)$

Is this a sufficient proof and correct use of proof by contrapositive?

Answer 1

However by definition, $a$ is a centre of $Z(U) \subset U$.

This isn't obvious: it's not going to be true for general subsets (For instance, an annulus is a subset of a disc, and while every point is a centre of a disc no points are centres of an annulus). In order to show that this is true for the specific subset $Z(U)$ it's not enough to have that the entire line $[a,b]$ lies in $U$ but also that every point on that line $[a,b]$ is also a centre.

Showing this is where the difficulty of this problem lies.

Hint: Try showing that any filled-in triangle, two of whose vertices are centres of $U$ and the third of which is any other point of $U$, lies in $U$.