On some 'moments' of planar convex bodies

We could generalize the concept of Moment of Inertia to a general moment thus: integral over the body of *some function* of the distance 'x' to a specified axis. If this function is x^2, we have the moment of inertia.

Question 1: For some specified angle α, can we always find some functional form of the moment such that thru every point of C, there is a pair of lines at angle α for which that moment of C is equal?

It is clear that for any planar convex region C and any moment, for any direction on the plane specified, there is a line that cuts C into 2 pieces with equal moment about that line. It is also obvious that if C is a circular disk, any line thru its center will cut it into two pieces of equal moment about itself, whatever be the functional form of the moment. We can call that line a 'moment bisector'.

Question 2: Under what conditions - shape of C and functional form of the moment - can a C without central symmetry have more than one moment bisector passing thru a point?
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Copying the content of this (Mathoverflow post ):
The following observation can be readily proved using the perpendicular axes theorem and intermediate value theorem: "Given any planar figure C, through any point on it, there is at least one pair of mutually perpendicular lines lying on the same plane such that the moments of inertia of C about the two lines are equal."

3. For a general planar convex figure C, what can one say about points on C with more than one such pair of mutually perpendicular lines? For example, is there only exactly one point with more than one pair of such mutually perpendicular lines through it (for a circular disk, its center obviously has infinitely many such line pairs)? Or is it that such a point can only be the center of symmetry if C is centrally symmetric?

4. Are there non-trivial variants of the above observation in 3D? For example, could one say: "for any convex 3D body C, there is at least one point P in it such that there is at least one set of 3 mutually perpendicular lines passing through P with moments of inertia of C about the 3 lines are equal."?

Aside: The perpendicular axes theorem seems to depend on the geometry being Euclidean. I don't know if it has an interesting variant in Hyperbolic geometry.
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Update: 10th december 2022: This mathoverflow post by Prof. Alexandre Ermenko reveals that question 4 from the mathoverflow post above has a "yes" answer.