The DNA Inequality in Non-Convex Regions

Abstract

A simple plane closed curve $Γ$ satisfies the DNA Inequality if the average curvature of any closed curve contained inside $Γ$ exceeds the average curvature of $Γ$. In 1997 Lagarias and Richardson proved that all convex curves satisfy the DNA Inequality and asked whether this is true for any non-convex curve. They conjectured that the DNA Inequality holds for certain L-shaped curves. In this paper, we disprove this conjecture for all L-Shapes and construct a large class of non-convex curves for which the DNA Inequality holds. We also give a polynomial-time procedure for determining whether any specific curve in a much larger class satisfies the DNA Inequality.

Versions 7--9 contains more figures, a summary of the proof, and other modifications. Version 6 has corrected a couple of minor notational problems with version 5. Versions 5--9 are (the same) major generalization of the theorem proved in versions 1--4