Motivated by the desire to cope with data imprecision , we study methods for preprocessing a set of line-segments (or just lines) in the plane such that whenever we are given a set of points, each of which lies on a distinct object, we can compute their convex hull more eficiently than in "standard settings" (that is, without preprocessing). In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P eficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(nα(n) log* n) expected time. If we further assume that the points are "oblivious" with respect to the data structure, the running time improves to O(nα(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-o- between space and query time and an output-sensitive algorithm. We also study the "dual problem" where we show how to efficiently compute the (≤k)-level of n lines in the plane, each of which lies on a distinct point (given in advance). We complement our results by Ω(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the "stan-dard" convex hull and sorting problems, in which the two problems require Θ(n log n) steps in the worst case (under the algebraic computation tree model).