Two pairs of centers for convex regions

Recording some experimental work done over the last week or so. A discussion page on this is at: https://math.stackexchange.com/questions/4482893/two-possible-triangle-centers

For every point P in a planar convex region C, we consider (1) the shortest and longest chords of C that pass thru P and (2) the shortest and longest distances from P to the boundary of C.

Define: 'Chord ratio(P)' = ratio between lengths of longest and shortest chords thru P. 'Chord difference(P)' = difference between lengths of longest and shortest chords thru P.

It is observed that when C is a triangle, both above quantities have a minimum value at the same interior point except for very flat triangles (this exceptional case was observed rather late so I was tricked into thinking that they are coincident for all triangles!). When C is a general convex region, the minima of both quantities are usually at different points in C.

We can also define: Distance ratio(P) as the ratio between longest and shortest distances to boundary of C from P and the Distance difference(P) as difference between these distances. It is seen that even when C is a triangle, these two quantities are minimized at different points in C.

Questions: Among all convex C's of unit diameter, which shape maximizes the distance between the minima of the distance ratio and distance difference? This question can be asked for the {chord ratio, chord difference} pair as well.

Note 1: More basically, when are these centers unique? Indeed, quantities defined above might have more than one minimum point for general C. For example, a square appears to have 4 distinct points in its interior that minimize the chord ratio.

Note 2: In 3D, one can define analogous quantities using (instead of chords) say, areas/perimeters of planar sections of C containing a point P inside C.

Cutting rectangles and squares into non-rectilinear tiles

A rectilinear polygon is one with all angles either 90 or 270 degrees. This post continues the overflow posts:
this and this

Question: Can a square (or even some rectangle) by tiled with an odd number N of mutually congruent polygons that are non-rectilinear? The tiles could be non-convex.

Note 1: If N is even, a square can be easily cut into say 4 mutually congruent and non-orthogonal non-convex pieces by 4 identical polylines that connect the center to the 4 vertices.

Note 2: this discussion is on (mostly) tiling rectangles with odd number of rectilinear tiles.

At the end is a question by Victor Protsak: "Do you know if there is a tiling of a rectangle into an odd number of congruent convex n-gons for n>=5?" That is pretty much what we asked above, except for relaxing convexity that is!

Non-congruent Tiling - 15

Here is part 14.
Question: For any positive real r1 and r2 with r2 < r1, there are infinitively many triangles with circumradius r1 and inradius r2. Is it possible to tile the plane with mutually non-congruent triangles all of same circumradius and inradius?

Note: I am not sure if we can tile the plane with (a) mutually non-congruent tiles all with same circumradius (b) with same inradius. This less constrained problem might have a "yes" answer.

This question has been put up at mathoverflow .

Further questions:
1. What are the invariants across triangles which have both circumradius and inradius same?
2. Do specifying the Steiner circumellipse and inellipse fully determine a triangle?
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A further mathoverflow post (July 3rd 2022) recorded these questions:

Given an ellipse E:

can we choose from those triangles for which E is the Steiner circumellipse, a pair-wise non-congruent set of triangles that tile the plane?

Can we choose from those triangles for which E is the Steiner inellipse, a pairwise non-congruent set of triangles that tile the plane?

Further query: As was noted in above linked page, given two positive real numbers r1 and r2 with r2 < r1/2. For any such pair, we observe that one can form infinitely many triangles all with circumradius r1 and inradius r2. Under what condition(s) will two specified ellipses be the Steiner circumellipse and inellipse of infinitely many triangles (or a unique triangle)? Basically, how do we characterize those triangles for which two specified ellipses are the Steiner ellipses?
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Let me also record a bunch of variants of the original fair partition question:

Question: Given a positive integer n, will every planar convex region allow partition into n convex pieces all with (a) same smallest containing circle (b) same largest inscribed circle and (c) both smallest containing circle and largest inscribed circle same? Partition into pieces with same largest inscribed circle looks plausible and easier by means of strips.

Here is what I gathered from Prof. Roman Karasev reg the status of the fair partition/spicy chicken problem:
First, the non-prime-power case of the spicy chicken theorem is so far only provable for the situation when one of the quantities being equalized over pieces is area (or something else continuous, zero on degenerate pieces, and additive) and the other value is continuous.
The prime power case works for some cases: If the two quantities we are trying to equalize are continuous and one of them is zero on degenerate pieces then a fair partition is also possible for prime power values of n.
Hence if you inscribe a circle or an ellipse, then its size and area are zero for a degenerate piece, which seems to do the job.