The Minimum Area of Convex Lattice n -Gons

The 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 [...

Finding 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 ...

Forced Convex n -Gons in the Plane

In a seminal paper from 1935, Erd}os and Szekeres showed that for each n there exists a least value g(n) such that any subset of g(n) points in the plane in general position must always contain the vertices of a convex n-gon. In particular, they obtained the bounds 2n 2 + 1 g(n) 2n 4 n 2 + 1 ; which have stood unchanged since then. In this note we remove the +1 from the upper bound for n 4. 1. ...

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...

The cost of cutting out convex n-gons