[math-fun] convex lattice 11-gon of minimum area?
Hello!
From Neil Sloane's sequences A063984 and A070911, I infer that the convex lattice 11-gon of minimum area is not yet known. This is remarkable, since the answer is known for 12-gons, 13-gons and 14-gons! Does anyone know about any recent progress toward a solution? Jamie Simpson proved in 1990 that the minimum area must be 19.5, 20, 20.5, 21 or 21.5.
By a convex lattice n-gon, I mean a polygon whose n vertices are points on the integer lattice Z^2 and whose interior angles are strictly less than pi. Thank you, Steve Finch http://pauillac.inria.fr/algo/bsolve/ _________________________________________________________________ Tired of slow downloads and busy signals? Get a high-speed Internet connection! Comparison-shop your local high-speed providers here. https://broadband.msn.com
At 11:37 AM 12/5/03 -0500, you wrote:
[... T]he convex lattice 11-gon of minimum area is not yet known. [... T]he minimum area must be 19.5, 20, 20.5, 21 or 21.5.
Just by pushing points around on graph paper, I was able to reconstruct a convex lattice 11-gon of area 43/2, the upper bound given by Simpson. This wasn't all that hard, actually, and made a pleasant puzzle which I recommend. There are a variety of helpful lemmas that one discovers, which I'm having trouble putting into clear words, about producing minimum-area n-sided caps with given "shoulders". -ACW
participants (2)
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Allan C. Wechsler -
Steven Finch