**Definitions:** The diameter of a convex region is the greatest distance between any pair of points in the region. The least width of a $2$D convex region can be defined as the least distance between any pair of parallel lines that touch the region. Let us refer to a polygon as 'fat' if the ratio between its diameter and least width (call this the 'thinness ratio') is low.

**General Question:** Given a general $n$, which $n$-gon is the fattest? For which values of $n$ (if at all) is the fattest n-gon a regular $n$-gon?

**2 Special cases:**
-For $n =3$, the equilateral triangle is the fattest. The thinness ratio is $2/\sqrt(3) \approx 1.15$.

-For $n = 4$, the square is not the fattest - its thinness ratio is $\sqrt{2}$. As pointed out by Yakov Baruch below, if we form a quad by adding a vertex arbitrarily close to one of the vertices of an equilateral triangle, its thinness ratio is only about $1.15$.

**Some Further Queries:** Thanks very much to Prof. O'Rourke for the pointer to Bezdek and Fodor's work (answer below) that answers just the above question. One could ask a further question: Given values for area and perimeter, which convex shape (not necessarily polygonal) is the most/least fat?

For fixed $A$ and $P$ within a suitable range (from perimeter equal to that of circle of area $A$ to perimeter of a Reuleux triangle with area $A$), it appears that fatness is maximized by curves of constant width (https://nandacumar.blogspot.com/2012/11/maximizing-and-minimizing-diameter-ii.html). For $P$ values greater than this range, I don't know.

For specified $A$ and $P$, diameter is maximized by a 'convex lens' shape as given in above linked page. But this shape does not seem to minimize fatness.