Castles in the air revisited

We show that the total number of faces bounding any one cell in an arrangement of n (d-1)-simplices in ℝ ^d is O(n ^d-1 log n), 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" is O(n ^d-1 log n). We, also show that the number of repetitions of a "k-flap," formed by intersecting d-k given simplices, along the boundary of the same cell, summed over all cells and all k-flaps, is O(n ^d-1 log² n). We use this quantity, which we call the excess of the arrangement, to derive bounds on the complexity of m distinct cells of such an arrangement.