An Upper Bound for Discrete Isometric Filling of Cycles

We study the discrete graph-metric analogue of Gromov's filling area problem for the cycle graph $C_n$. An abstract triangulation $K$ is an isometric filling of $C_n$ if $\partial K=C_n$ and the graph distance between any two boundary vertices is not shortened inside the $1$-skeleton of $K$. Let $D(n;\epsilon)$ denote the minimum number of vertices in a $(1-\epsilon)$-Lipschitz filling of $C_n$, and set $$D^*=\liminf_{\epsilon\to0^+}\liminf_{n\to\infty}\frac{D(n;\epsilon)}{n^2}.$$ Previous work gives the general lower bound $D^*\ge 1/8$, while discretizing the hemisphere gives the upper bound $$D^*\le \frac{1}{\pi\sqrt3}.$$ In this paper we give an explicit discrete construction which improves the hemispherical upper bound. More precisely, we construct isometric fillings $K_n$ of $C_n$ with $$|V(K_n)|\le \left(\frac16+o(1)\right)n^2,$$ and hence $$D^*\le \frac16<\frac{1}{\pi\sqrt3}.$$ This can directly illustrate the discrete filling area problem is a proper relaxation of Gromov's original filling area problem and cannot be used to settle Gromov's conjecture. The construction is a concentric annular filling.