More On Oriented Containers and 'Containees'

A few years ago, I wrote this doc: https://arxiv.org/abs/1802.10447

Elaborating on one of the questions introduced therein (to find least area isosceles triangle that contains a given triangle), Kiss, Pach and Somlai proved some serious results here:

https://arxiv.org/pdf/2001.09525.pdf

Here is a bunch of questions (which have algorithmic as well as classical aspects) in the same vein:

Given a polygon (especially, a triangle), to find/characterize the:

- least area (or least perimeter) centrally symmetric polygon that contains it.
- max area (or max perimeter) centrally symmetric polygon that is contained within it.
- the least area(least perimeter) axisymmetric polygon (a polygon with at least one direction of symmetry) that contains it.
- max area(max perimeter) axisymmetric polygon that is contained within it.

Note 1: The concept of 'contained region with max perimeter' (whether axisymmetric or full centrally symmetric) can have a definite answer only if we are constrained to search only for convex regions contained within the input region; even with that constraint, the answer might be degenerate.

Note 2: I am not sure if there is an O(n) algorithm to check if a given n-gon is axisymmetric. An O(n^2) algorithm is easy to show: one just needs to check 2*n candidate lines - the n angular bisectors and the n perpendicular bisctors of the sides; each line can be tested in O(n) time to decide whether it is an axis of symmetry. Note: A discussion on this particular question is here: https://mathoverflow.net/questions/387005/check-if-a-polygon-has-an-axis-of-symmetry

Guesses:
- At least for some triangles, the least area axisymmetric container appears to be non-convex. Iam not sure if these optimal axisymmetric containers of triangles can be got by deforming(trimming) smallest isosceles containers (or special isosceles triangle containers - definition of 'special' is given in the paper by Kiss, Pach and Somlai mentioned above) of triangles.

- For any convex region, the max area contained region (whether axisymmetric or rotationally symmetric) is probably convex.