The Minimum Area of Convex Lattice n-Gons
نویسندگان
چکیده
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 [8] via an inequality of Rényi–Sulanke [9]
منابع مشابه
New Algorithms for Minimum Area k-gons
Given a set P of n points in the plane, we wish to find a set Q ⊂ P of k points for which the convex hull conv(Q) has the minimum area. We solve this, and the related problem of finding a minimum area convex k-gon, in time O(n log n) and space O(n log n) for fixed k, almost matching known bounds for the minimum area triangle problem. Our algorithm is based on finding a certain number of nearest...
متن کاملDowker-type theorems for hyperconvex discs
A hyperconvex disc of radius r is a planar set with nonempty interior that is the intersection of closed circular discs of radius r. A convex disc-polygon of radius r is a set with nonempty interior that is the intersection of a finite number of closed circular discs of radius r. We prove that the maximum area and perimeter of convex disc-n-gons of radius r contained in a hyperconvex disc of ra...
متن کاملThe minimum area of a simple polygon with given side lengths
Answering a question of H. Harborth, for any given a1, ..., an > 0, satisfying ai < ∑ j 6=i aj we determine the infimum of the areas of the simple n–gons in the Euclidean plane, having sides of length a1, ..., an (in some order). The infimum is attained (in limit) if the polygon degenerates into a certain kind of triangle, plus some parts of zero area. We show the same result for simple polygon...
متن کاملFinding Minimum Area k-gons
Given a set P of n points in the plane and a number k, we want to find a polygon ~ with vertices in P of minimum area that satisfies one of the following properties: (1) cK is a convex k-gon, (2) ~ is an empty convex k-gon, or (3) ~ is the convex hull of exactly k points of P. We give algorithms for solving each of these three problems in time O(kn3). The space complexity is O(n) for k = 4 and ...
متن کامل4-Holes in point sets
We consider a variant of a question of Erdős on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n ≥ 9, the maximum number of general 4-holes is ( n 4 ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Combinatorica
دوره 24 شماره
صفحات -
تاریخ انتشار 2004