An Upper Bound for Discrete Isometric Filling of Cycles

Abstract

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.