The DNA Inequality in Non-Convex Regions

Details The DNA Inequality in Non-Convex Regions The DNA Inequality in Non-Convex Regions

2008-01-13

arxiv.org/abs/0801.1929v9

Mathematics

Metric Geometry

Versions 7--9 contains more figures, a summary of the proof, and other modifications. Version 6 has corrected a couple of mino

Scientific paper

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.

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