Three Questions...

Recording 3 questions recently posted at MathOverflow:

1. On Geometry of Numbers:

https://mathoverflow.net/questions/381542/on-circles-and-ellipses-drawn-on-an-infinite-planar-square-lattice

Thanks to Profs. Noam Elkies and Alexei Ustinov for sharing their insights.

Just recording the main questions: Consider a square lattice on the plane formed by points with integer coordinates:

- Given any positive integer n, can we always find a sufficiently large circle drawn on the plane that passes through at least n lattice points? Can such circles be found if the center is not to be a lattice point? What if we require the circle to pass through exactly n lattice points?

- Above qn has a natural restatement if instead of circles, we look at ellipses (either all with a given eccentricity e or with e that can be freely chosen). The ellipses need not be axis parallel.

- And what can one say if the lattice of points has as unit cell not a square but a general parallelogram? And what happens in 3D and higher dimensions?

2. On Zonogons:

https://mathoverflow.net/questions/381781/partitions-of-convex-planar-regions-into-zonogons

A discussion is on...

Here are the main questions: A zonogon is a centrally symmetric convex polygon.

- Are there convex non-zonogons that can be partitioned into a finite number of (convex) zonogons?

- Same as 1 with the pieces allowed to be nonconvex but centrally symmetric polygons.

3. On Congruent Partitions:

https://mathoverflow.net/questions/381091/on-congruent-partitions-of-planar-regions

Given any integer n, any rectangular region or any sector of a disc (including the full disk as a boundary case) can be cut into n mutually congruent pieces - by equally spaced parallel lines and lines radiating from a point at equal angular spacing respectively.

Intuitively, this property generalizes to some deformations of the rectangle which continue to allow partition into n congruent pieces by mutually parallel and equally spaced polylines or curves for any n and for some deformations of a sector which can be congruent partitioned by mutually congruent curves radiating from a single point. .

Question: Are there any other classes of planar regions which can be cut into n mutually congruent regions for any n? The answer seems negative, but is t

here a proof? Note: Analogously, in 3D, one readily has parallelopipeds and suitable slices of regions with axial symmetry (sphere, torus, cone...) which can obviously be cut into n mutually congruent 3D regions for any n.

Fair Partitions and Voronoi Vertices

 The proof of the Fair Partition/ Spicy Chicken problem begins with weighted Voronoi decompositions of planar convex regions. The friendliest intro to this would be https://www.youtube.com/watch?v=5SfXqTENV_Q&t=11s

The basic thought behind this post: What if we do not allow the Voronoi vertices to have different weights? This is of course, the basic partition of a convex region into Voronoi regions based on a sprinkle of vertices. 

With unweighted Voronoi diagrams we can partition any given convex region C into n convex pieces of equal area (only equal area). Indeed, when the centers are on the same line then the Voronoi parts are strips (planks). Adjusting the centers you may obtain arbitrary widths of the strips. Then using the Knaster--Kuratowski--Mazurkiewicz theorem you may also obtain equal area intersections of the strips with $C$. (this explanation is due to Prof. Roman Karasev).

Further thought:  Is it possible to find a set of n vertices within C that are seeds of an 'unweighted' Voronoi decomposition of C into n convex pieces all of same area and perimeter is achieved? Or maybe a partition so that a finite set of quantities {area, perimeter...} are equal across pieces? For most convex regions, for most n values, there may be no such voronoi vertex sets at all.

Suspicion: One has the feeling that if a set of n interior points exists within C yielding an unweighted voronoi partition into pieces with equal area and perimeter (and maybe a few more such properties) then this point set contains important information about the structure of C.

And further one guesses: if for even some values of n, such a set of points has to exist inside C, then C has to have some special symmetry properties (if such a 'voronoi set' of points exists for all values of n, C might be v special). One might even have some theorem that if a small set of global properties are to be equal over n pieces and a set of unweighted Voronoi vertices exists wihin C to achieve this, then, C might have sufficient symmetry to necessarily require that the pieces will necessarily be congruent to another.