The DNA Inequality states that the average curvature of a curve inside of a given closed figure exceeds the average curvature of the figure. In the paper by Lagarias and Richardson (1997) that proved it for convex figures, the question arose if it could be possible to prove it for some non-convex region; the authors suggested L-Shaped regions. In this paper, we disprove the conjecture for L-Sha...

We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms maximal number rooks that can be placed polyomino. also characterize Gorenstein polyominoes and those which are on punctured spectrum, compute Cohen–Macaulay type any polyomino rectangles covering it. Though main results algebraic nature, all proofs combinatorial.