There is a convex shape $C$. It is known that the largest disc contained in $C$ has radius $r$ and the smallest disc containing $C$ has radius $R$. What are the smallest-area and largest-area shapes that fit this description?

My current conjecture is based on the following diagram:

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For the largest-area shape, I tried to build the widest shape that is still not wider than the enclosed disc. This is the shape QTUS (where again the lines QT and US are curved). In this case, the measured area is slightly less than $4rR$, and it seems to converge to $4rR$ when $R\gg r$. This makes sense because QTUS converges to a rectangle whose side-lengths are the diameters of the two discs ($2r\cdot 2R$).

For the smallest-area shape, I tried to build the most "economic" shape that still contains a full diameter of the enclosing disc. This is the shpae CGJDIH (where the lines GJ and IH are curved around the disc). The area of this shape changes when the enclosed disc moves along the diameter; the minimal area seems to be when the two discs have the same center. In this case, the empirically-measured area is slightly more than $2rR$, and it seems to converge to $2rR$ when $R\gg r$. This makes ssense because CGJDIH converges to a union of two triangles and the area is 1/2 the area of QTUS.

MY QUESTION IS: are these indeed the smallest and largest possible convex shapes?


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