Castles in the air revisited

Abstract

We show that the total number of faces bounding any one cell in an arrangement ofn (d−1)-simplices in ℝd isO(n d−1 logn), thus almost settling a conjecture of Pach and Sharir. We present several applications of this result, mainly to translational motion planning in polyhedral environments. We than extend our analysis to derive other results on complexity in arrangements of simplices. For example, we show that in such an arrangement the total number of vertices incident to the same cell on more than one “side” isO(n d−1 logn). We, also show that the number of repetitions of a “k-flap,” formed by intersectingd−k given simplices, along the boundary of the same cell, summed over all cells and allk-flaps, isO(n d−1 log2 n). We use this quantity, which we call theexcess of the arrangement, to derive bounds on the complexity ofm distinct cells of such an arrangement.