منابع مشابه
On convex lattice polygons
Let II be a convex lattice polygon with b boundary points and c (5 1) interior points. We show that for any given a , the number b satisfies b 5 2e + 7 , and identify the polygons for which equality holds. A lattice polygon II is a simple polygon whose vertices are points of the integral lattice. We let A = 4(11) denote the area of II , b{U) the number of lattice points on the boundary of II , ...
متن کاملConvex lattice polygons of fixed area with perimeter-dependent weights.
We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight tm to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s varies as s(-theta(conv))eK(t)square root(s) for large s and t less than a critical threshold tc, where K(t) is a t-dependent constant, and theta(conv) is a ...
متن کاملConvex Lattice Polygons
Let n ≥ 3 be an integer. A convex lattice n-gon is a polygon whose n vertices are points on the integer lattice Z 2 and whose interior angles are strictly less than π. Let a n denote the least possible area enclosed by a convex lattice n-gon, then [1, 2, 3] {a n } ∞ n=3 = n 1 2
متن کاملThe Minimum Area of Convex Lattice n-Gons
What is the minimal area A(n) a convex lattice polygon with n vertices can have? The first to answer this question was G.E. Andrews [1]. He proved that A(n)≥cn3 with some universal constant c. V.I. Arnol’d arrived to the same question from another direction [2], and proved the same estimate. Further proofs are due to W. Schmidt [10], Bárány–Pach [3]. The best lower bound comes form Rabinowitz [...
متن کاملAsymptotic behaviour of convex and column-convex lattice polygons with fixed area and varying perimeter
We study the inflated phase of two dimensional lattice polygons, both convex and column-convex, with fixed area A and variable perimeter, when a weight μ exp[−Jb] is associated to a polygon with perimeter t and b bends. The mean perimeter is calculated as a function of the fugacity μ and the bending rigidity J . In the limit μ → 0, the mean perimeter has the asymptotic behaviour 〈t〉/4 √ A ≃ 1−K...
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ژورنال
عنوان ژورنال: Bulletin of the Australian Mathematical Society
سال: 1990
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972700028525