Given a set of points S={p1,. . ., pn} in Euclidean d -dimensional space, we address the problem of computing the d -dimensional annulus of smallest width containing the set. We give a complete characterization of the centers of annuli which are locally minimal in arbitrary dimension and we show that, for d=2 , a locally minimal annulus has two points on the inner circle and two points on the outer circle that interlace anglewise as seen from the center of the annulus. Using this characterization, we show that, given a circular order of the points, there is at most one locally minimal annulus consistent with that order and it can be computed in time O(n log n) using a simple algorithm. Furthermore, when points are in convex position, the problem can be solved in optimal Θ(n) time.

Abstract In this paper we consider polygonizations that are robust when faced with changes in the vertices that are present or in their position. We analyze the dynamic maintenance of different types of polygonizations (monotone, star‐shaped…) and we introduce monotone half‐convex polygonizations that are specially interesting because they provide minimum cost per insertion or deletion. If we had to delete not only one point but several external layers of the set, then the onion polygonizations would be suited, because they can be updated in constant time. We also consider the case of points that can be moved to contiguous positions and we show how to polygonize the set for updating in linear time. We deal too with security problems for a polygon: What is the maximum distance the vertices of a polygon could be moved away of their position in such a way that the topology on the boundary of the polygon (or its convexity) remains the same?.