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It is known (see for example the Wikipedia entry on the Reuleaux triangle) that for every curve of constant width (CCW), the largest inscribed circle and the smallest circumscribed circle are concentric, and their radii sum to the constant width of the curve. So among CCWs with unit width, the same curve both minimizes the largest inscribed circle and maximizes the smallest circumscribed circle. Guess: the Reuleaux triangle is the CCW that achieves it.

Question: What can one say about largest area (perimeter) inscribed and smallest area(perimeter) circumscribed ellipses of CCWs of unit width? For example, one can ask: in a CCW, how different in orientation can be largest area inscribed ellipse and least area circumscribed ellipse - or the largest area and largest perimeter inscribed ellipses...?

Indeed, there are several obvious possible sub-questions. I don't know of any known results.

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