On some moments of Planar convex bodies - 3

Copying questions as recorded at Mathoverflow:
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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.

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?

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.

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 x2) equal between the two pieces?
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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 M1 and M2 of a convex lamina C about two lines that are at some specified angle α, 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 M1, M2 and α. In particular, one could ask: if M1=M2 and α 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 Mx, My and Mz 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 x2).

Isosceles Triangle Containers - an inside-out variant

We continue from this earlier post
Ref:
1. https://arxiv.org/abs/2001.09525
2. https://arxiv.org/abs/2205.11637

The above two papers address the questions of finding/characterizing the smallest enclosing and largest embedded isosceles triangles of a given general triangle (Note: 'smallest' and 'largest' may be defined with respect to either area and perimeter).

Let us record the inside-out versions of the same questions:
Given any isosceles triangle T to find that smallest triangle for which T is the smallest containing isosceles triangle. Likewise, find the largest triangle for which T is the largest contained(embedded) isosceles triangle.

The guess is that the perimeter versions of this question could be more difficult as opposed to the area versions.
Note: One can pose the same question with 'isosceles' replaced with 'right'. And of course, all these 'oriented container' investigations could have interesting variants in hyperbolic geometry.

One can generalize the question thus: Given an isosceles triangle T, to find the smallest(largest) convex region for which T is the smallest containing(largest embedded) isosceles triangle.

One could also ask for specific pairs (general triangle, isosceles triangle) that maximize the difference between container and containee.

A somewhat related overflow discussion
Updates to follow...

On some moments of planar convex bodies - 2

Let us begin with a question that is under discussion. Thanks to Professors Roman Karasev and Arseniy Akopyan

1. An ellipse is usually defined through the pair of its foci lying on its major axis. Is there a natural way to define an ellipse through a pair of points lying on its minor axis?
Note: We mean real points. In particular, there must be two points on the minor axis of the ellipse such that the inertia form (the second moment) of the solid ellipse about those points is round (is a scalar matrix).
Of course, it may well happen that this analytic definition does not correspond to anything geometric. And let me add, this question got deleted from mathoverflow for reasons I don't understand.
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2. Let us define a moment bisector to be a line l thru a planar region C that cuts it into two pieces with equal moment (of inertia or defined in any way given in last post here) about l itself.
Question: What can we say about points on C that have more than one moment bisector thru it?

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.