Exercise 12, § 2

A subset E of $\mathbf{R}^n$ which contains the origin o is said to be starlike (with respect to o) if, for all $x \in E$ and all $t \in [0, 1]$, we have $tx \in E$. The intersection of E with a closed ray of origin o is either the whole ray or a segment of which o is one end-point. The shell of E is the set K consisting of the non-zero end-points of these segments, together with o if there exists a ray whose intersection with E consists of o alone. In what follows we shall assume that $o \notin K$.

a) Show that the shell of E is contained in the frontier of E. Give an example in which these two sets are different.

b) Show that, if the shell K of E is compact, there is a homeomorphism of $\mathbf{R}^n$ onto itself which maps K onto $S_{n-1}$, $\overline{E}$ onto $B_n$ and the interior of E onto the interior of $B_n$ (map each point $x \in K$ to its central projection on $S_{n-1}$, and then extend this mapping to the whole of $\mathbf{R}^n$). Deduce that the frontier of E coincides with its shell, and that the interior of E is the set of points $tx$, where $x \in K$ and $t \in [0, 1[$.

c) If E is unbounded and if its frontier coincides with its shell, then the interior of E is homeomorphic to $\mathbf{R}^n$, and its shell K is homeomorphic to an open subset of $S_{n-1}$ [show that the image of E under the homeomorphism $x \to x/(1 + ||x||)$ satisfies the conditions of b)].

d) Give an example of an unbounded starlike set E whose shell is closed but is not identical with the frontier of E.