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Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry)

Consider an alternative definition of axiality as follows: For a convex region C, consider a chord L and the set of chords of C that are perpendicular to L. For each perpendicular chord, consider the ratio: length of smaller segment to length of larger segment into which L divides it. Now, for chord L, define an 'axiality ratio' as the minimum of these ratios over all its perpendicular chords. Now, the best axis of C is that chord for which axiality ratio is a maximum and that value of the axiality ratio could be called the axiality of C itself.

Question: What shape of C minimizes axiality under this new definition?

Note: Some more related thoughts are recorded at http://nandacumar.blogspot.com/2023/01/axiality-of-planar-convex-regions.html and http://nandacumar.blogspot.com/2023/01/centralness-of-convex-planar-regions.html

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Too many bodies have vanishing axiality ratio. Indeed, if the ratio for $L$ is positive, then the whole body projects to $L$. So you may start with convex smooth body, locate these exceptional chords and modify the body in a small neighborhood of its ends making the ratio = zero.

In particular, you get a body with zero ratio arbitrarily close to any given body (in the sense of Hausdorff).

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  • $\begingroup$ Thanks for pointing out a serious problem with the definition - especially body C not projecting onto chord L. The following approach too appears to fail to capture the extent of axial symmetry: consider for each chord L, the minimum among all chords perpendicular to L of the ratio: (larger segment length / smaller seg length), and then, take the minimum over all chords of this axiality ratio as the axiality of C itself. $\endgroup$ Jan 14, 2023 at 9:39

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