We had a lengthy discussion yesterday, on how to prove that the circumference of a circle is $2\pi r$. By using google, the most commonly found proof starts in the following way.
"Consider the regular n-gon inside the circle touching the circle at its vertices, and the regular n-gon outside the circle touching the circle at its edges. Then the circumference of the inner n-gon is smaller than the circumference of the circle, which is smaller than the circumference of the outer n-gon."
While this apparently is true for n-gons, it is not true for arbitrary geometric shapes inside and outside the circle. So we conjectured it was the convexity of the n-gon that is of importance. Which leads to the question:
Let $C$ be a circle with radius $r$, and $A_n$ be a family of convex sets with $A_n \subset C$ for each $n$. If, for $n \rightarrow \infty$, the volume of these sets converges to $\pi r^2$ (the volume of $C$), will the circumference necessarily converge to $2\pi r$?