The DNA Inequality in Non-Convex Regions
  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. 
Eric Larson