Abstract
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 some 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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