A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please visit here .
My question is:
Suppose that $E\subset\mathbb R^n$ is open, bounded, and starlike, and $p_0$ is a center for $E$.
(a):Is it true or false that all points $p_1$ in a small enough neighborhood of $p_0$ are also centers for $E$ ?
(b):Can it consist of a single point?
Refrence: Real Mathematical Analysis-Charles Chapman Pugh .
(a) false; (b) true. I give an example in two dimensions; you can generalize it for $n>2$.
$$E = \{(x,y) : |x|<2, |y|<2\}\setminus \left[\{(x,0): |x|\ge 1\}\cup \{(0,y): |y|\ge 1\} \right] $$
The only center of $E$ is $(0,0)$.