By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C.

Question: For specified area and perimeter which C maximizes moment of inertia? Which shape of C minimizes it?

Note 1: One could ask variants with say, "specified area, perimeter and diameter" and so on.

Note 2: The same questions can be asked for other moments (moment of inertia is defined for squares of distances from axis; other moments for other powers).

Note 3: Relaxing the requirement of convexity of C but with all other conditions fixed could have interesting implications for moments defined with respect to higher powers of the distance to axis.

Earlier questions: To find the convex planar region minimizing diameter when area and perimeter are given, Which $n$-gons of diameter 1 maximize the moment of inertia?

By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C.

Question: For specified area and perimeter which C maximizes moment of inertia? Which shape of C minimizes it?

Note 1: One could ask variants with say, "specified area, perimeter and diameter" and so on.

Note 2: The same questions can be asked for other moments (moment of inertia is defined for square of distance from axis; other moments for other powers).

Note 3: Relaxing the requirement of convexity of C but with all other conditions fixed could have interesting implications for moments defined with respect to higher powers of the distance to axis.

Earlier questions: To find the convex planar region minimizing diameter when area and perimeter are given, Which $n$-gons of diameter 1 maximize the moment of inertia?