This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia.
The perpendicular axis theorem states that the moment of inertia of a planar lamina about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia of the lamina about the two axes at right angles to each other, in its own plane intersecting each other at the point where the perpendicular axis passes through it.
If we are given the moments of inertia $M_1$ and $M_2$ of a convex lamina C about two lines that are at some specified angle $\alpha$, what could be said about the moment of inertia about an axis perpendicular to the plane through the point of intersection of the two lines? The MI about this perpendicular axis might lie in some range determined by $M_1$, $M_2$ and $\alpha$. In particular, one could ask: if $M_1$=$M_2$ and $\alpha$ is specified, which shape of C maximizes (minimizes) the MI of C about the perpendicular axis?
Moving to 3D, if the moments of inertia of a convex body about the 3 axes $M_x$, $M_y$ and $M_z$ are given, how closely could one calculate the MI of the body about some other line through the origin?
Note: One can also consider integrals of other functions of the distance x to the line (other than $x^2$).
