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I was always puzzled by this fact.

A uniform cube has a sphere-shaped inertia ellipsoid. The sphere has a higher symmetry then the cube.

Is there any deep reason or implication behind it?

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The inertia ellipsoid is computed from an integral about an axis - in other words you rotate the object. This will "smooth out" any symmetries and typically increase the symmetry.

Sorry this is a "early morning" intuitive explanation - maybe someone else will give you a more formal answer.

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It is a good answer indeed. – ja72 Jul 20 '14 at 15:48

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