Extremal problems on convex lattice polygons in sense of \(l_p\)-metrics
type: Item
instance of: scholarly article
title: Extremal problems on convex lattice polygons in sense of \(l_p\)-metrics
zbMATH Open document ID: 1022.52006
DOI: 10.1016/S0012-365X(02)00384-9
published in: Discrete Mathematics
publication date: 2003-03-16T00:00:00Z
review text: This paper presents (and proves) the following asymptotic expression for the minimal \(l_{p}\)-perimeter \(s_{p}(n)\) of a convex lattice polygon with \(n\) vertices: \(s_{p}(n)=c n^{3/2}+O(n)\) with \(c=2\pi /\sqrt{54 A_{p}}\), where \(A_{p}\) is the area of the \(l_{p}\) unit disc \(\{(x,y)\in \mathbb{R}^{2}:\left|x\right|^{p}+\left|y\right|^{p}\leq 1\}\), and \(p>1\) is an integer. Conversely, for the maximal number \(N_{p}(s)\) of vertices of a convex lattice polygon with \(l_{p}\)-perimeter \(s\) the formula \(N_{p}(s)=c^{\prime } s^{2/3}+O(s^{1/3})\) with \(c^{\prime }=1/c^{2/3}\) is obtained. In the cases \(p=1\) and \(p=\infty \) the rest terms have to be replaced by \(O(n\log n)\) and \(O(s^{1/3}\log s)\), respectively.
Mathematics Subject Classification ID: 52C05
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 1882153
zbMATH Keywords: lattice polygons
zbMATH Keywords: minimal \(l_{p}\)-perimeter
Wikidata QID: Q57942652
author: Joviša Žunić
reviewed by: Johann Linhart
MaRDI profile type: Publication
full work available at URL: s0012-365x(02)00384-9
OpenAlex ID: W2121790088
DBLP publication ID: journals/dm/Zunic02
description: scientific article; zbMATH DE number 1882153