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Background: Among convex plane $n$-gons of unit diameter, we can try to achieve:

  • the largest area. (This is called the biggest little polygon with $n$ sides; for $n$ odd, the regular polygon on $n$ sides is known to be the biggest little $n$-gon; for $n=6$ the regular polygon is suboptimal and Ron Graham found an optimal polygon; for $n=8$ the regular polygon is again suboptimal; for higher even $n$ I am not sure).
  • the largest width or the largest perimeter. (The Reinhardt polygons maximize both of these.)
  • the largest moment of inertia about an axis through the center of mass and normal to the plane (our present question).

Question: Among convex $n$-gons of diameter 1, which maximize this moment of inertia?

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  • $\begingroup$ Perhaps someone can provide formulas for the moments of some regular polygons, Reinhardt polygons, and biggest little polygons? That would be a good place to start. $\endgroup$
    – user44143
    Commented Feb 4, 2021 at 17:43
  • $\begingroup$ The recent article of math.toronto.edu/mccann/papers/LimMcCann20.pdf contains results on such variational questions. $\endgroup$
    – JHM
    Commented Feb 4, 2021 at 22:04

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