Data imprecision constitutes an important gap between theory and practice in computational geometry. A lot of research about imprecision in computational geometry is directed at computing the convex hull of imprecise points rather than imprecise line segment intersection. In this paper we introduce an algorithm to construct the convex hull for a set of imprecise line segment intersection in time.

The Shapley-Folkman theorem and its corollaries [ 1, 2, 3, 4, 5, 6, 81 provide strong bounds on the distance between the sum of a family of nonconvex sets and the convex hull of the sum. Proofs of the theorem are nonconstructive, and require moderately advanced analysis. The proof developed below is based on elementary considerations. It provides an approximation sequentially with the successiv...

In this note, we characterize the convex hull of the Stiefel manifold and we find its strong convexity parameter. We also introduce the notion of roundness of a set and show that the Stiefel manifold is round.

Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles a...

One classical result of Freiman gives the optimal lower bound for the cardinality of A + A if A is a d-dimensional finite set in R. Matolcsi and Ruzsa have recently generalized this lower bound to |A+ kB| if B is d-dimensional, and A is contained in the convex hull of B. We characterize the equality case of the Matolcsi–Ruzsa bound. The argument is based partially on understanding triangulation...

A set of vertices C in a graph is convex if it contains all vertices which lie on shortest paths between vertices in C. The convex hull of a set of vertices S is the smallest convex set containing S. The hull number h(G) of a graph G is the smallest cardinality of a set of vertices whose convex hull is the vertex set of G. For a connected triangle-free graph G of order n and diameter d ≥ 3, we ...

In this paper, a representative algorithm convex hull with half-dividing and recurrence is commented; and according to the isomorphic fundamental theorem of the convex hull construction, and guided by the isomorphic distributing characteristics of a convex hull’s the apexes, a more efficient new algorithm to find a convex hull based on the dynamical base line with a maximum pitch of the dynamic...