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We consider uniform convex planar regions and lines through their center of mass and lying in the same plane as the region; each line is parametrized by an angle $\alpha$ it makes with some reference direction in that same plane.

Are there uniformly dense convex planar regions other than the circular disk with the property: the region has equal moment of inertia about every line through the center of mass and lying within a finite range of values of $\alpha$? For the circular disk, this range is obviously, the full [0,2$\pi$].

Further question: And what could be said about other moments?

Note added on 16th November 2023: The question has been answered below for the case of moment of inertia; I guess there could be issues regarding other moments - defined in terms of powers other than quadratic of distance from an axis.

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    $\begingroup$ What do you mean by "uniform " in "uniform convex planar regions"? Also, what do you mean, formally, by "a finite range"? $\endgroup$
    – Iosif Pinelis
    Commented Nov 5, 2023 at 15:22
  • $\begingroup$ by uniform, I meant uniformly dense. We need the moment of inertia of the convex region to be the same with respect to all lines with orientation ranging from some alpha_1 to alpha_2 (that is different from alpha_1). $\endgroup$
    – Nandakumar R
    Commented Nov 5, 2023 at 16:04

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The moment of inertia is a quadratic form: for a region $R$ with respect to a line through the origin in the direction of a unit vector $v$, it is given by $\int_{R}(v\cdot x)^2\,dx=(Av)\cdot v$ for a symmetric, positive definite matrix $A$ called inertia tensor. If the eigenvectors $v_{1,2}$ of $A$ are distinct, then $(Av)\cdot v$ is strictly monotone on each of the four arcs of the unit circle between $\pm v_{1,2}$. Therefore, under you condition, moments of inertia with respect to all lines through the origin are in fact equal.

There are plenty of regions with this property, for example, any regular polygon will do, since for an $n$-gon, the eigenvector $v_{1}$ with a larger eigenvalue rotated by $\frac{2\pi}{n}$ is again an eigenvector with the same eigenvalue, which is only possible if $A$ is a scalar matrix.

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  • $\begingroup$ Thanks. I understand that any regular polygon is just as good as the circular disk - they all have same moment of inertia about any line through the center. From your answer it also appears that there is no planar convex region for which the moment of inertia is same only for lines that lie within some subinterval of the full [0,2*pi] orientations. Hope to have your take on moments other than quadratic too. marking the question as answered. $\endgroup$
    – Nandakumar R
    Commented Nov 8, 2023 at 6:21

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