Approximation of the Willmore energy by a discrete geometry model

We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of Γ-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature.

Further steps on the reconstruction of convex polyominoes from orthogonal projections

A remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

Review of Forbidden Configurations in Discrete Geometry by David Eppstein

In 1930, the mathematician Esther Klein observed that any five points in the plane in general position (i.e., no three points forming a line) contain four points forming a convex quadrilateral. This innocentsounding discovery led to major lines of research in discrete geometry. Klein's friends Paul Erdős and George Szekeres generalized this theorem, and also conjectured that 2k-2 + 1 points (again in general position) would be enough to force a convex k-gon to exist. The resolution of this conjecture became known as the "happy ending problem," because Klein and Szekeres ended up getting married. The unhappy side is that it has, to date, not been completely solved, although a recent breakthrough of Suk made significant progress. This both mathematically and personally charming little story is a great beginning for this elegant book about discrete geometry. It typifies the type of problems that are studied throughout, and also captures the spirit of curiosity that drives such studies. The book covers many problems that lie at the intersection of three fields: discrete geometry, algorithms and computational complexity.