We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two nonoverlapping convex polygons in the plane. Given two non-overlapping convex polygons P1 and P2 in the plane, the ham-sandwich cut of P1 and P2 is a line that simultaneously bisects the area (or perimeter or vertex count) of both polygons. We provide a data structure that supports queries for the ham-s...

2 Abstract. We present a new method that allows to enumerate various classes of column-convex polygons, according to their perimeter and their area. The rst step of this method leads to a functional equation which deenes implicitly the generating function of the class under consideration. The second step consists in solving this equation. We apply systematically our method to all the usual clas...

Given n points in R, we give approximation algorithms to find a subset of k of the points that has minimum-area or minimum-perimeter convex hull. We give algorithms that individually yield constantfactor approximations to the minimum-perimeter problem with different running times for differing values of k. Together, these algorithms provide a constant-factor approximation of the minimumperimete...

Hexagonal polyominoes are polyominoes on the honeycomb lattice. We enumerate the symmetry classes of convex hexagonal polyominoes. Here convexity is to be understood as convexity along the three main column directions. We deduce the generating series of free (i.e. up to reflection and rotation) and of asymmetric convex hexagonal polyominoes, according to area and half-perimeter. We give explici...

We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter 〈L〉 and the mean area 〈A〉 have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), li...

Hexagonal polyominoes are polyominoes on the honeycomb lattice. We enumerate the symmetry classes of convex hexagonal polyominoes. Here convexity is to be understood as convexity along the three main column directions. We deduce the generating series of free (i.e. up to reflection and rotation) and of asymmetric convex hexagonal polyominoes, according to area and half-perimeter. We give explici...

Every convex set in the plane gives rise to geometric functionals such as the area, perimeter, diameter, width, inradius and circumradius. In this paper, we prove new inequalities involving these geometric functionals for planar convex sets containing zero or one interior lattice point. We also conjecture two results concerning sets containing one interior lattice point. Finally, we summarize k...