Jarńık’s convex lattice n-gon for non-symmetric norms
نویسندگان
چکیده
What is the minimum perimeter of a convex lattice n-gon? This question was answered by Jarńık in 1926. We solve the same question in the case when perimeter is measured by a (not necessarily symmetric) norm.
منابع مشابه
Jarník’s convex lattice n-gon for non-symmetric norms
What is the minimum perimeter of a convex lattice n-gon? This question was answered by Jarník in 1926. We solve the same question, and prove a limit shape result, in the case when perimeter is measured by a (not necessarily symmetric) norm.
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 Unit Distance Problem for Centrally Symmetric Convex Polygons
Let f(n) be the maximum number of unit distances determined by the vertices of a convex n-gon. Erdős and Moser conjectured that this function is linear. Supporting this conjecture we prove that f (n) ∼ 2n where f (n) is the restriction of f (n) to centrally symmetric convex n-gons. We also present two applications of this result. Given a strictly convex domain K with smooth boundary, if fK (n) ...
متن کاملTamari lattices and noncrossing partitions in type B and D
The usual, or type An, Tamari lattice is a partial order on T A n , the triangulations of an (n+3)-gon. We define a partial order on T B n , the set of centrally symmetric triangulations of a (2n + 2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the An Tamari lattice, and therefore that it deserves to be considered the Bn Tamari lattice. We define a bij...
متن کاملOn symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon
Motivated by the question of the polytopal realizability of the simplicial complex Γn,k of (k + 1)-crossing-free sets of diagonals of the convex n-gon, we study the first non-obvious case, namely the simplicial complex Γ8,2 of 3-crossing-free sets of diagonals of the convex octagon. We give a complete description of the space of symmetric realizations of Γ8,2, that is, of the polytopes P whose ...
متن کامل