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Numerator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911.
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%I #37 Nov 18 2022 09:23:28

%S 2,2,50,8,10,10,1250,29,40,52,73,73,82,82,23290,148,202,226,317,317,

%T 365,452,500,530

%N Numerator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911.

%C Without the minimal area stipulation, the result differs for some n. (See n = 12 in the examples.) - _Peter Munn_, Nov 17 2022

%H Hugo Pfoertner, <a href="/A321693/a321693_1.pdf">Illustrations of optimal polygons for n <= 26</a> (2018).

%e For n = 5, the polygon with minimal area A070911(5) = 5 and enclosing circle of least diameter is

%e 2 D

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%e | + +

%e | + +

%e 1 E C

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%e | + +

%e | + +

%e 0 A + + + B

%e 0 ----- 1 ----- 2 ---

%e .

%e The enclosing circle passes through points A (0,0), C (2,1) and D (1,2). Its diameter is sqrt(50/9). Therefore a(5) = 50 and A322029(5) = 9.

%e For n = 11, a strictly convex polygon ABCDEFGHIJKA with minimal area and enclosing circle of least diameter is

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

%e 5 J ++++++ I

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%e | + +

%e 4 K . H

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%e | + . +

%e | + +

%e 3 A . +

%e | + . +

%e | + . . +

%e | + . +

%e 2 B O G

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%e | + . +

%e | + . +

%e 1 C F

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%e | + +

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%e 0 D ++++++ E

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

%e .

%e The diameter d of the enclosing circle is determined by points A and F, with I also lying on this circle. d^2 = 6^2 + 2^2 = 40. Therefore a(11) = 40 and A322029(11) = 1.

%e n = 12 is a case where the minimal area stipulation is significant. If we take the upper 6 edges in the n = 11 illustration above and rotate them about the enclosing circle's center to generate another 6 edges, we get a 12-gon with relevant squared diameter a(11) = 40 that meets all criteria except minimal area. This 12-gon's area is 26, and to meet the minimal area A070911(12)/2 = 24, the least squared diameter achievable is 52 (see illustration in the Pfoertner link). So a(12) = 52 and A322029(12) = 1. - _Peter Munn_, Nov 17 2022

%Y Cf. A070911, A192493, A192494, A322029 (corresponding denominators).

%K nonn,frac,hard

%O 3,1

%A _Hugo Pfoertner_, Nov 21 2018

%E a(21)-a(26) from _Hugo Pfoertner_, Dec 03 2018