Limit behaviour of constant distance boundaries of Jordan curves

For a Jordan curve Gamma in the complex plane, its constant distance boundary Gamma(lambda) is an inflated version of Gamma. A flatness condition, (1/2, r(0))-chordal property, guarantees that Gamma(lambda) is a Jordan curve when lambda is not too large. We prove that Gamma(lambda) converges to Gamma, as lambda approaching to 0, in the sense of Hausdorff distance if Gamma has the (1/2, r(0))-chordal property.

Automated summary

This paper investigates Jordan curves and their constant distance boundaries in the complex plane, proving the convergence of Gamma(lambda) to Gamma under certain conditions. This has important implications for understanding the properties of Jordan curves.