Three problems about dynamic convex hulls

We present three results related to dynamic convex hulls: A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O(log^1+ε n) for an arbitrarily small constant ε > 0. This improves the previous bound of O(log^3/2 n). A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O(log n + k) time with O(polylog n) expected amortized update time. A similar result holds for 3-dimensional orthogonal range reporting. For 3- dimensional halfspace range reporting, the query time increases to O(log² n/ log log n + k). A semi-online dynamic data structure for maintaining a set of n line segments in the plane, so that we can decide whether a query line segment lies completely above the lower envelope, with query time O(log n) and amortized update time O(n^ε). As a corollary, we can solve the following problem in O(n^1+ε) time: given a triangulated terrain in 3-d of size n, identify all faces that are partially visible from a fixed viewpoint.