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>



On 'axiality' of planar convex regions
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).


