[seqfan] Re: Enclosing Circle of Convex Lattice Polygons

[Brad Klee]

Hi Hugo,

Nice idea, and the drawings [1] really help a lot more than the ASCII
art ( maybe you should show all of the bounding circles though? ) .

It seems like for n=8*m, there is some resonance with the underlying
square geometry, and possibly dihedral symmetry will hold throughout.

To create a corner the slope must change, and convexity seems to
imply a regular structure for the delta sequence across the horizontal.

For example:

n=8   :   0 + 1 + 1 + 1 + 0 = 3
n=16 :   0 + 1 + 1 + 2 + 1 + 2 + 1 + 1 + 0 = 9
n=24 :   0 + 1 + 1 + 1 + 2 + 3 + 1 + 3 + 2 + 1 + 1 + 1 + 0 = 17

And I can guess that

n=32 :   0 + 1 + 1 + 1 + 1 + 2 + 3 + 4 + 1 + 4 + 3 + 2 + 1 + 1 + 1 + 1 + 0 = 27
n=m*8:  2*m - 1 +  (m+1)*m = m^2 + 3*m - 1 .

As m->oo and n->oo, these summations should converge to the diameter
length. If the limit indeed exists, then it should be perfectly fine
to calculate
its value on a subsequence.

Cheers,

Brad

[1] https://oeis.org/A322106/a322106.pdf

On Mon, Dec 10, 2018 at 3:55 PM Hugo Pfoertner <yae9911 at gmail.com> wrote:
>
> SeqFans,
>
> during the last weeks I've performed computations to extend
> https://oeis.org/A063984 and the equivalent https://oeis.org/A070911 . For
> polygons with larger n typically more than one solution minimizing the
> polygon area exists, and I found it interesting to find the polygons of
> minimal area fitting into the smallest enclosing circle. The corresponding
> results are in https://oeis.org/A321693 / https://oeis.org/A322029
>
> A related problem is to find the smallest circle into which at least one
> suitably shaped convex lattice n-gon can be inscribed. For n<12 the
> solutions are identical to the minimum area problem, but starting at n=12
> polygons leading to a minimum diameter can have larger enclosed areas than
> the minimum area n-gons.
>
> My question is now about the expected asymptotic behavior of the squared
> diameters of the smallest possible enclosing circles. There is a "frac"
> pair of sequences https://oeis.org/draft/A322106 / https://oeis.org/A322107
> with my results. Based on the empirical results showing a tendency of the
> optimal n-gons towards a circular shape I had conjectured that
>
> Lim_{n->oo} (A322106(n)/ A322107(n)) / n^2 = 1/2
>
> but at the moment I have removed this from the draft. Any ideas how to
> prove or disprove this conjecture are welcome.
>
> Hugo Pfoertner
>
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