TITLE: Suppose $S$ is bounded star shaped subset of $ \mathbb R^n$ where $n \ge 2$ then $ \partial S$ is path connected
QUESTION [5 upvotes]: Def: A set $S$ in the Euclidean space $ \mathbb R^n $ is called a star shaped domain if there exists $x_o$ in $S$ such that for all $x$ in $S$ the line segment from $x_0$ to $x$ is in $S$.
Some days ago in a comment in the answer to one of my questions @Hagen Von Eitzen commented that  

Suppose $S$ is bounded  star shaped subset of $ \mathbb R^n$ where $n \ge 2$ then $ \partial S$ is Path connected.

I am unable to prove this result. Please help!

REPLY [6 votes]: Here's another example. Let $(r,\phi)$ be the polar coordinates in $\mathbb R^2$. Define $S$ as the closure of $$\left\{(r,\phi)\mid r\leq2+\sin\frac{30}{\phi(2\pi-\phi)}\right\},$$ where we only allow $r\geq 0$ and $\phi\in(0,2\pi)$. See here for a picture.
This is star shaped pretty much by definition (since for each $\phi$ it contains all the points of at most a certain distance from the origin) and its boundary is a variant of the Warsaw circle and is not path connected.