On The First Unknown Value of Two Functions For Convex Lattice $v$-Gons

Abstract

Let $a(v)$ and $g(v)$ denote the least possible area and the least possible number of lattice points in the interior of a convex lattice $v$-gon, respectively. Many lower and upper bounds for $a(v)$ and $g(v)$ are known for every $v$. However, the exact values of these two functions are only known for $v\le 10$ and $v\in \{12,13,14,16,18,20,22\}$. The purpose of this paper is to answer the following Open Question 1 from $[13]$: What is the exact value of $a(11)$? We answer this question by proving that $a(11)=21.5$. On our way to achieve this goal, we also prove that $g(11)=17$.