How can it be proved that for all star-shaped regions bounded by a surface given parametically by:
$\textbf{r}(t) = (a_1,....,a_n)+\textbf{p}(t), t\in [a,b]$
where $(a_1,...,a_n)$ is the point the region is star-shaped about,
the set of all points on the surface or on the interior of the surface are given by:
$\textbf{k}(t) = (a_1,...,a_n) + \frac{\textbf{p}(t)}{\vert\textbf{p}(t)\vert}k,\;\;k\in [0,\vert \textbf{p}(t)\vert],\;\; t\in [a,b]$
$K = \overline{A} = A+\partial A$
Have to show:
$(x_1,....,x_n) \in K$
$\Updownarrow$
$(x_1,...,x_n) = (a_1,...,a_n) + \frac{\textbf{p}(t)}{\vert\textbf{p}(t)\vert}k,\;\;k\in [0,\vert \textbf{p}(t)\vert],\;\; t\in [a,b]$
1.
Since A is bounded a vector from the point $\textbf{c}$ it is star-shaped about to the boundary $\partial A$ can be created for all points on the boundary.
Since the region is star-shaped about this point all points on the vector, exluding the end-point if A is open, will be contained in A. So all points given by $\textbf{k}(t)$ are contained in K.
2.
Since the region is bounded a line drawn through $\textbf{c}$ and an arbitrary point $\textbf{p}$ in A will at some point cross $\partial A$. Call this point $\textbf{r}_l$.
Therefore a vector can be created from $\textbf{c}$ to $\textbf{r}_l$ which will contain $\textbf{p}$.
All points on this vector will be given parammetrically by:
$(x_1,....,x_n) = \textbf{c} + k\frac{\textbf{r}_l-\textbf{c}}{\vert\textbf{r}_l-\textbf{c}\vert}, k \in [0, \vert\textbf{r}_l-\textbf{c}\vert]$.
Since all points on $\partial A$ are described by $\textbf{r}(t), t\in[a,b]$ there exists a number $o \in [a,b]$ such that $\textbf{r}(o) = \textbf{r}_l$. Hence,
$(x_1,....,x_n) =(a_1,...,a_n) +k\frac{\textbf{p}(o)}{\vert\textbf{p}(o)\vert}, k \in [0,\vert \textbf{p}(o)\vert] = \textbf{k}(o)$
Since all points on this vector are described by this, so is our point. Therefore the point is described by $\textbf{k}(t)$