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 also 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 (my question here).

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

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?

Background: The biggest little polygon with $n$ sides is the convex plane $n$-gon of unit polygon diameter having largest possible area. For $n$ odd, the regular polygon on $n$ sides is known to be the biggest little $n$-gon. For even $n$, the regular polygon is suboptimal for $n$=6 (this optimal hexagon was found by Ron Graham) and $n$ = 8 (not sure about higher even numbers). The Reinhardt polygons are polygons maximizing perimeter for fixed diameter, maximizing width for their diameter, and maximizing width for their perimeter.

Question: If we try to keep diameter at 1 and try to find n-gons that maximize the moment of inertia about an axis thru the center of mass and normal to the plane, what happens?

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 also 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 (my question here).

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