I have a shape thus:
In Illustrator, this design is composed of a load of paths with a heavy stroke applied, and no fill. How can I obtain the path describing the white area?
Note: I drew this in Illustrator but coloured it in Photoshop for clarity
Use Object > Expand to turn the paths into fills, then use Pathfinder operations as required to produce a single shape for the white area.
Honestly, I would cheat.
1) You already have it in Photoshop, right? Use the Magic Wand tool to select the white area. Under Paths, click on Work Path and save it as Path 1. Export that back to Illustrator.
Open your original and lock it. Open the new exported path and copy that to the locked original (to use as a trace). Then tweak the curves on your new path until they match the original.
2) Or alternatively, it's a very simple shape... lock the original and redraw it with the Pen tool. Should take you all of five minutes.