Ref: Mathematical Omnibus by Fuchs and Tabachnikov, Lecture 11.

Consider any planar convex region C. A line l may be called an inertia bisector of C if it divides C into 2 pieces each of which has same moment of inertia with respect to l.

1. Which shape of C causes the envelope of all its inertia bisectors to enclose the maximum fraction of the area of C? Guess: a triangle. But will any triangle do?
2. If for a certain C, all inertia bisectors are concurrent, is C necessarily centrally symmetric? One would guess it is.

Note: As in reference given above, one can ask about lines that cut C into two pieces with their moments of inertia in some specified ratio rather than equal to each other.

3. What happens if instead of the moment of inertia one tries to make the integral of some other function of the distance x to the line (other than x$^2$) equal between the two pieces?

Ref: Mathematical Omnibus by Fuchs and Tabachnikov, Lecture 11.

Consider any planar convex region C. A line l on the same plane and cutting thru C may be called an inertia bisector of C if it divides C into 2 pieces each of which has same moment of inertia with respect to l.

1. Which shape of C causes the envelope of all its inertia bisectors to enclose the maximum fraction of the area of C? Guess: a triangle. But will any triangle do?
2. If for a certain C, all inertia bisectors are concurrent, is C necessarily centrally symmetric? One would guess it is.

Note: As in reference given above, one can ask about lines that cut C into two pieces with their moments of inertia in some specified ratio rather than equal to each other.

3. What happens if instead of the moment of inertia one tries to make the integral of some other function of the distance x to the line (other than x$^2$) equal between the two pieces?