Extremal problems on convex lattice polygons in sense of \(l_p\)-metrics
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.
scientific article; zbMATH DE number 1882153