On 'Deployment' and 'Dispersal'

A basic intro to these classes of problems is given in Stanley Ogilvy's 'Tomorrow's Math'

Deployment: To place a specified number n of station points in a region such that the maximum distance of any point in the region from one of the stations is minimized.

Dispersal: To place n station poits in a region such that the minimum distance among the stations is maximized.

Example: It is not possible to scatter (disperse) 5 points on a sphere so that every pair of them is separated by more than 90 degrees. The same statement is true for 6 points. For dispersal 5 points and 6 points give same result.

Question: What is the status of dispersal on a triangular region?

Remark 1: For n =2, it is best to place the stations at the ends of the longest side. For any higher n, it appears that 2 of the stations should be kept at the ends of the longest side. For n =3, the third station is either (a) the third vertex of the triangle or (b) the point where the perpendicular bisector of the longest side cuts the rest of the triangle. Beyond that, things are less clear.

Remark 2: Reg best deployment of n points on a triangular region, even for n = 2, things are far from obvious. Wonder if there are any special properties to the Voronoi regions that can result when n stations are optimally deployed - whether the regions will have say equal area.

Discussion page on this: https://mathoverflow.net/questions/388977/deployment-and-dispersion-in-triangular-regions

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.

3 Questions Posted at Overflow

Just for the record, storing the links:
1. Fattest Polygons based on Diameter and Least Width:
https://mathoverflow.net/questions/385856/fattest-polygons-based-on-diameter-and-least-width

2. Convex Lattice Polygons of Equal Area and Perimeter:
https://mathoverflow.net/questions/386399/convex-lattice-polygons-with-equal-area-and-perimeter

3. On One-Parameter Families of curves Drawn on a Lattice Background:
https://mathoverflow.net/questions/386647/on-one-parameter-families-of-curves-drawn-on-a-2d-lattice-background