On the last few years multidimensional projection techniques have advanced towards defining faster and user-centered approaches. However, most of existing methods are designed as generic tools without considering particular features of the data under processing, such as the distance distribution when the data is embedded into a certain metric space. In this paper we split the projection techniques into two groups, global and local techniques, conduct an analysis of them, and present a novel local technique specially designed for projecting heavy tail distance distributions, such as the one produced by high-dimensional sparse spaces. This novel approach, called Local Convex Hull (LoCH), relies on an iterative process that seeks to place each point close to the convex hull of its nearest neighbors. The accuracy, in terms of neighborhood preservation, is confirmed by a set of comparisons and tests, showing that LoCH is capable of successfully segregating groups of similar instances embedded in high-dimensional sparse spaces and of defining the borders between them, significantly better than most projection techniques.