Fair cake-cutting when players join late
Assume the cake/area is a circle $C$ with radius $r$. Then, we can create 
$n$ fair pieces by the canonical cake cutting and thus assign each player an area
of $\frac{\pi r^2}{n}$. We can then assign the $(n+1)$th a small circle around the center, 
and subsequent new players rings around that one (swallowing part of the inner circle), 
and so on. In this way, every player gets one contiguous piece/plot which shrinks 
in a monotone way for the first $n+1$ players, and moves towards the center for 
those joining later.

Working out the numbers is simple; for the first new player, simply solve

$\qquad \pi r_1^2 = \frac{\pi r^2}{n+1}$

to get the radius for his plot. For the second, solve

$\qquad\begin{align}
  \pi r_1^2 &amp;= \frac{\pi r^2}{n+2} \\
  \pi r_2^2 - \pi r_1^2 &amp;= \frac{\pi r^2}{n+2}
\end{align}$

to get (outer) radii for both new players, and so on. 
It seems that player $n + i$ gets outer radius $r_i = \frac{r\sqrt{i}}{\sqrt{n+k}}$
after $k$ additional players have joined, but I did not prove this.

This is what you get for $n=6$ and $k=0,1,2,3$:


[source]

The same idea works for regular polygons with $n$ sides. If you assume a rectangle
as basic form, you can do a similar thing by assigning the first $n$ equally sized
columns and the following players rows (starting at one side).