An Upper Bound for Discrete Isometric Filling of Cycles

TL;DR

This paper improves the upper bound for the discrete isometric filling of cycle graphs by constructing a concentric annular filling with fewer vertices, refining previous bounds and illustrating the problem's relaxation of Gromov's original conjecture.

Contribution

The authors provide an explicit discrete construction that reduces the upper bound for the filling area problem, surpassing the hemispherical discretization bound.

Findings

Constructed isometric fillings with |V(K_n)| ≤ (1/6 + o(1)) n^2

Established D* ≤ 1/6, improving previous upper bounds

Demonstrated the discrete problem is a proper relaxation of Gromov's conjecture

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.