Kites

Kites are a class of 'oriented convex polygons' which I didn't know about earlier. The latest post here on oriented containers is is: http://nandacumar.blogspot.com/2021/03/more-on-oriented-containers-and.html

In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other instead of being adjacent. (wiki)

Kiss, Pach and Somlai have published results on the smallest area isosceles triangle that contains a given triangle (https://arxiv.org/abs/2001.09525) and I understand results on the smallest perimeter isosceles containers of a given general triangle have also been derived.

Question: Intiutively, Kites are to general quadrilaterals what isosceles triangles are to general triangles. So, one can naturally ask if there are interesting constraints on the least area (least perimeter) containing kites of a given general convex quadrialteral - like whether the smallest kite container necessarily shares a corner with the quad. And what about the smallest kite containing a given convex n-gon?

Some further thoughts and questions:

- Is there an upper bound on the difference in orientation between the smallest area and smallest perimeter containing (and contained) kites for a convex polygon - and if so which convex polygon achieves it? Thus there are obviously, algorithmic questions on finding the smallest containing and contained kites for a given convex polygon.
- It is clear that any triangle can be cut into 3 kites that meet at the incenter. So, a general n-gon can be cut into 3*n kites. Is this a tight lower bound?
Further, like kites, one can wonder about partitioning n-gons into cyclic polygons.

In case I didn't record this question earlier, let me for completeness, note the questions of finding the largest contained (smallest containing) central symmetric region (axisymmetric region) for a given polygon. At the moment, I have no good ideas! A seemingly related question is here A question on paritioning into zonogons (centrally symmetric polygons) was discussed here .

And I dont know any convex planar region without central symmetry that allows partition into some finite number of centrally symmetric convex (or even non convex!) planar regions.

It is clear that an n-gon (convex or otherwise) can be cut into O(n) triangles and thence into 3n kites. Is this a tight lower bound on th eminimum number of pieces needed if an n- gon is to be cut ibto axisymmetic pieces?

Constant Width Curves - Packing

A quote by H L Resnikoff: https://arxiv.org/pdf/1504.06733.pdf

I conjecture that the packing by Reuleaux triangles has the greatest density for packings of the plane by a curve of constant width. Moreover, it seems likely that, given two curves of the same constant width, the one with the smaller area has the greater maximal packing density. The reason is that the curve that bounds the smaller area will be ‘less round’ and offer more opportunities to snuggle closer to its neighbors.

The above conjecture seems to imply that the circle has least density of packing among constant width curves. The construction given by Martin Gardner (Colossal Book of Math, page 38) for constant width curves with unequal arcs puts a bit of doubt on this. Indeed, I am not sure if the worst packer of the plane among convex regions is necessarily a curve with central symmetry (the present best candidate with central symmetry is the 'smoothed octagon').