Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs
Proceedings of the twenty-fourth annual symposium on Computational geometry
Lower bounds for weak epsilon-nets and stair-convexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
| Hi-index | 0.00 |
A compact set $S \subset {\Bbb R}^2$ is staircase connected if every two points $a,b \in S$ can be connected by a polygonal path with sides parallel to the coordinate axes, which is both x-monotone and y-monotone. $\xi(a,b)$ denotes the smallest number of edges of such a path. $\xi(\cdot,\cdot)$ is an integer-valued metric on S. We investigate this metric and introduce stars and kernels. Our main result is that the r-th kernel is nonempty, compact and staircase connected provided $r \ge \frac{1}{2} \cdot {\it stdiam}(S) +1$.