Calin Popescu: On the area and the lattice diameter of lattice triangles, 59-62

Abstract:

Given an integer $n \ge 2$, let $f(n)$ be the largest area a lattice triangle of lattice diameter at most $n$ may have. We prove that, if $n \ge 4$, then $f(n) \ge \frac 12 (n^2 + 3)$, and $f(n) \ge \frac{19}{32} n^2 > \frac 12 (n^2 + 3)$ for infinitely many $n$.

As a corollary, given any non-negative integer $N$, the largest possible area of a lattice triangle of lattice diameter $n$ is greater than $\frac{19}{32} (n + N)^2$ for infinitely many $n$.

Key Words: Lattice diameter, lattice triangles.

2020 Mathematics Subject Classification: Primary 52C05; Secondary 11H06.

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