  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.
