Definition 7.4 (Convex and star-shaped domains):

(i) A subset $X$ of $\mathbb{R}^n$ is said to be convex if for every pair of points $a$ and $b$ in $X$, the line segment $ta+(1-t)b$, $0 \leq t \leq 1$ lies entirely in $X$.

(ii) A subset X of $\mathbb{R}^n$ is said to be star shaped with respect to a point $x_0$ if for every $a\in X$, the line segment $ta+(1-t)x_0$, $0 \leq t \leq 1$ lies entirely in $X$.

So a convex domain is star shaped with respect to any of its points. in