scholarly journals Convex lattice polygons of minimum area

1990 ◽  
Vol 42 (3) ◽  
pp. 353-367 ◽  
Author(s):  
R.J. Simpson
Keyword(s):  
Integer Lattice ◽  
Minimum Area ◽  
Lattice Polygon ◽  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Lattice Polygons

A convex lattice polygon is a polygon whose vertices are points on the integer lattice and whose interior angles are strictly less than π radians. We define a(2n) to be the least possible area of a convex lattice polygon with 2n vertices. A method for constructing convex lattice polygons with area a(2n) is described, and values of a(2n) for low n are obtained.

scholarly journals A note on bounds on the minimum area of convex lattice polygons

1992 ◽  
Vol 45 (2) ◽  
pp. 237-240 ◽  
Author(s):  
Charles J. Colbourn ◽  
R.J. Simpson
Keyword(s):  
Positive Constant ◽  
Minimum Area ◽  
Lattice Polygon ◽  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Lattice Polygons

The minimum area a(v) of a v–sided convex lattice polygon is known to satisfy . We conjecture that a(v) = cv3 + o(v3), for c a constant; we prove that , and that for some positive constant c, .

scholarly journals On convex lattice polygons

1976 ◽  
Vol 15 (3) ◽  
pp. 395-399 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Boundary Points ◽  
Lattice Polygon ◽  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Interior Points ◽  
Lattice Polygons

Let π be a convex lattice polygon with b boundary points and c (≥ 1) interior points. We show that for any given c, the number b satisfies b ≤ 2c + 7, and identify the polygons for which equality holds.

scholarly journals The Perimeter of optimal convex lattice polygons in the sense of different metrics

2001 ◽  
Vol 63 (2) ◽  
pp. 229-242 ◽  
Author(s):  
Miloš Stojaković
Keyword(s):  
Asymptotic Expression ◽  
Planar Shape ◽  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Lattice Polygons

Classes of convex lattice polygons which have minimal lp-perimeter with respect to the number of their vertices are said to be optimal in the sense of the lp-metric.It is proved that if p and q are arbitrary integers or ∞, the asymptotic expression for the lq-perimeter of these optimal convex lattice polygons Qp(n) as a function of the number of their vertices n is . for arbitrary ɛ > 0, where . and Ap is equal to the area of the planar shape |x|p + |y|p ≤ 1.

scholarly journals On the Number of Convex Lattice Polygons

1992 ◽  
Vol 1 (4) ◽  
pp. 295-302 ◽  
Author(s):  
Imre Bárány ◽  
János Pach
Keyword(s):  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Lattice Polygons

We prove that there are at most {cA1/3} different lattice polygons of area A. This improves a result of V. I. Arnol'd.

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