I'll take the chance to mention the following nice [paper](http://arxiv.org/pdf/1010.4611.pdf) of Aronov and Hubard.  In it they answer a question of Nandakumar and Rao of whether a convex body can be partitioned into $n$ equal pieces of equal volume *and* equal perimeter.  Their proof shows that this is always possible (in any dimension) when $n$ is prime.  

I particularly like the following real-world application (taken directly from their abstract).  

>Imagine that you are cooking chicken at a party. You will cut the raw chicken fillet with a
sharp knife, marinate each of the pieces in a spicy sauce and then fry the pieces. The surface
of each piece will be crispy and spicy. Can you cut the chicken so that all your guests get the
same amount of crispy crust and the same amount of chicken?