[seqfan] Enclosing Circle of Convex Lattice Polygons

Hugo Pfoertner yae9911 at gmail.com
Mon Dec 10 22:54:42 CET 2018


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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