The DNA Inequality in Non-Convex Regions

TL;DR

This paper investigates the DNA Inequality for non-convex regions, disproves a previous conjecture for L-shaped curves, and provides a polynomial-time method to verify the inequality for a broad class of curves.

Contribution

It disproves the conjecture that all L-shaped curves satisfy the DNA Inequality and introduces an efficient procedure to check the inequality for many non-convex curves.

Findings

Disproved the DNA Inequality for all L-shaped curves.

Constructed a large class of non-convex curves satisfying the inequality.

Provided a polynomial-time algorithm to verify the inequality for specific curves.

Abstract

A simple plane closed curve $\Gamma$ satisfies the DNA Inequality if the average curvature of any closed curve contained inside $\Gamma$ exceeds the average curvature of $\Gamma$. 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.