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Numerator of the least possible squared diameter of an enclosing circle of a strictly convex lattice n-gon.
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%I #29 Oct 19 2022 06:42:48

%S 2,2,50,8,10,10,1250,29,40,40,2738,72,82,82,176900,17810,1709690,178,

%T 11300,260,290,290,568690,416,2418050,488,3479450,629,2674061,730

%N Numerator of the least possible squared diameter of an enclosing circle of a strictly convex lattice n-gon.

%C If the smallest possible enclosing circle is essentially determined by 3 vertices of the polygon, the squared diameter may be rational and thus A322107(n) > 1.

%C The first difference of the sequences A321693(n) / A322029(n) from a(n) / A322107(n) occurs for n = 12.

%C The ratio (A321693(n)/A322029(n)) / (a(n)/A322107(n)) will grow for larger n due to the tendency of the minimum area polygons to approach elliptical shapes with increasing aspect ratio, whereas the polygons leading to small enclosing circles will approach circular shape.

%C For n>=19, polygons with different areas may fit into the enclosing circle of minimal diameter. See examples in pdf at Pfoertner link.

%D See A063984.

%H Hugo Pfoertner, <a href="/A322106/a322106.pdf">Illustration of convex n-gons fitting into smallest circle</a>, (2018).

%H Hugo Pfoertner, <a href="/A322106/a322106_1.pdf">Illustration of convex n-gons fitting into smallest circle, n = 27..32</a>, (2018).

%e By n-gon a convex lattice n-gon is meant, area is understood omitting the factor 1/2. The following picture shows a comparison between the minimum area polygon and the polygon fitting in the smallest possible enclosing circle for n=12:

%e .

%e 0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6

%e 6 H ##### Gxh +++++ g

%e | # + # * +

%e | # + # +

%e | # + * # +

%e 5 I i F f

%e | # + * # +

%e | # + # +

%e | # + * # +

%e 4 J j # e

%e | # @+ * # +

%e | # + @ #+

%e | # + @ * +#

%e 3 K + @ + E

%e | # + * @ + #

%e | # @ + #

%e | + # * +@ #

%e 2 k # d D

%e | + # * + #

%e | + # + #

%e | + # * + #

%e 1 l L c C

%e | + # * + #

%e | + # + #

%e | + * # + #

%e 0 a ++++ Axb ##### B

%e 0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6

%e .

%e The 12-gon ABCDEFGHIJKLA with area 52 fits into a circle of squared diameter 40, e.g. determined by the distance D - J, indicated by @@@. No convex 12-gon with a smaller enclosing circle exists. Therefore a(n) = 40 and A322107(12) = 1.

%e For comparison, the 12-gon abcdefghijkla with minimal area A070911(12) = 48 requires a larger enclosing circle with squared diameter A321693(12)/A322029(12) = 52/1, e.g. determined by the distance a - g, indicated by ***.

%Y Cf. A063984, A070911, A321693, A322029, A322107 (corresponding denominators).

%K nonn,frac,more

%O 3,1

%A _Hugo Pfoertner_, Nov 26 2018

%E a(27)-a(32) from _Hugo Pfoertner_, Dec 19 2018