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.

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    $\begingroup$ I feel that you have already provided a good answer to your question by your $f(P)$. Could you please indicate why you're unsatisfied by that answer. $\endgroup$ – André Henriques Nov 22 '14 at 18:21
  • $\begingroup$ Well, okay I have defined $f(P)$. Now I would like to know if I can say something about it. $\endgroup$ – seub Nov 22 '14 at 20:21
  • $\begingroup$ If the island lies in a hemisphere then the minimal disc is uniquely defined (it is easy to proof). $\endgroup$ – Anton Petrunin Nov 22 '14 at 20:36
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    $\begingroup$ One possible generalization would be to take the convex hull of your domain, rather than the bounding disk/rectangle. $\endgroup$ – Jeremy Salwen Nov 23 '14 at 0:30
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    $\begingroup$ A notion of a fat region is used in computational geometry: e.g., On fat partitioning, fat covering etc. $\endgroup$ – Joseph O'Rourke Nov 23 '14 at 2:54

Similar issues come up in studying gerrymandering (drawing political districts with partisan objectives), where it's useful to have a measure of how "irregular" a region is.

You can read about various classical irregularity measures in this political science paper: Measuring the Compactness of Legislative Districts (Young, Legislative Studies Quarterly, 1988). The author calls your measure the "Roeck Test".

For more mathematically-oriented references, see the answers to this math.stackexchange question.

EDIT: You asked about computability, so it's worth mentioning that there's a linear time algorithm to find the smallest circle containing a plane region.

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