A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as large as possible? He had something in mind like: if

$$\frac{\mathrm{area}(\mathrm{island})}{\mathrm{area}(\mathrm{water} + \mathrm{island})} < 50\%$$

then the island is "thin", otherwise the island is "fat". I thought about it a little bit but could not come up with anything enlightening.

Here is a mathematical translation: let $P$ be a bounded domain in the plane (I suppose all you need is $P$ to be a measurable set). Define the "fatness" constant $$f(P) = \max_D \frac{\mathrm{area}(P)}{\mathrm{area}(D)}$$ where the max is taken over all Euclidean disks $D$ containing $P$.

I don't have a very precise question, I just wonder: is there anything interesting to be said about $f$? Has it been studied? Does it have notable properties? Is it computable in some cases? etc.

Note that this "fatness constant" could be generalized in several ways. For starters one could take Euclidean rectangles instead of disks. Another example of generalization would be to ask the question on the round sphere rather than in the plane. Here is another idea that comes to my mind: what if the island lives on the sphere but you want to draw it in the plane? Could we ask that same question over all representations of the island that are conformal (or satisfy some other property)?

NB: As has been pointed out, the words "thin" and "fat" I used here are probably a poor choice of terminology, but I did not have a better idea.

fat regionis used in computational geometry: e.g., On fat partitioning, fat covering etc. $\endgroup$ – Joseph O'Rourke Nov 23 '14 at 2:54