We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two overlapping point sets in convex position in the plane. The ham-sandwich cut of S1 and S2 is a line that simultaneously bisects the area, perimeter or vertex count of both point sets. Our algorithm supports insertion and deletion of vertices in O(log n) time, and area, perimeter and vertex-count queries...

We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two (possibly overlapping) point sets in the plane, with a bounded number of convex-hull peeling layers. The ham-sandwich cut of S1 and S2 is a line that simultaneously bisects the area, perimeter or vertex count of both point sets. Our algorithm supports insertion and deletion of vertices in O(c log n) tim...

An important class of planar straight-line drawings of graphs are the convex drawings, in which all faces are drawn as convex polygons. A graph is said to be convex planar if it admits a convex drawing. We consider the problem of testing convex planarity in a semidynamic environment, where a graph is subject to on-line insertions of vertices and edges. We present on-line algorithms for convex p...

We present new results for three problems dealing with a set P of n convex constantcomplexity fat polyhedra in 3-space. (i) We describe a data structure for vertical ray shooting in P that has O(log n) query time and uses O(n log n) storage. (ii) We give an algorithm to compute in O(n log n) time a depth order on P if it exists. (iii) We give an algorithm to verify in O(n log n) time whether a ...

We prove several results of the following type: given finite dimensional normed space V there exists another space X with log dimX = O(log dimV ) and such that every subspace (or quotient) of X, whose dimension is not “too small,” contains a further subspace isometric to V . This sheds new light on the structure of such large subspaces or quotients (resp., large sections or projections of conve...