In this paper, an effective method with time complexity of O(K3/2N2 log K ǫ0 ) is introduced to find an approximation of the convex hull for N points in dimension n, where K is close to the number of vertices of the approximation. Since the time complexity is independent of dimension, this method is highly suitable for the data in high dimensions. Utilizing a greedy approach, the proposed metho...

The dot comparison task, in which participants select the more numerous of two dot arrays, has become the predominant method of assessing Approximate Number System (ANS) acuity. Creation of the dot arrays requires the manipulation of visual characteristics, such as dot size and convex hull. For the task to provide a valid measure of ANS acuity, participants must ignore these characteristics and...

Let K be a smooth convex set with volume one in R. Choose n random points in K independently according to the uniform distribution. The convex hull of these points, denoted by Kn, is called a random polytope. We prove that several key functionals of Kn satisfy the central limit theorem as n tends to infinity.

In this note, we prove the following result. Let K ⊂ Rd be a convex body with the origin O in its interior. If there is a number λ ∈ (0, 1) such that the n-dimensional volume of the convex hull of the union of K with the translates of λK , by a vector x , depends only on the Euclidean norm of x , then K is a Euclidean ball.

Radon’s theorem asserts that any set S of d + 2 points in [Wd has a partition into two subsets S, and S, such that Conv(S,) rl Conv(S,) # 0, where Conv(Si) denotes the convex hull of Si. This central theorem in the theory of convexity has been extended in many directions. One of the most interesting generalizations is the following theorem, proved by Tverberg in 1966, and now considered as a cl...

The evenly convex hull of a given set is the intersection of all the open halfspaces which contain such set (hence the convex hull is contained in the evenly convex hull). This paper deals with nite dimensional linear systems containing strict inequalities and (possibly) weak inequalities as well as equalities. The number of inequalities and equalities in these systems is arbitrary (possibly in...

We explore one method for finding the convex hull of certain mixed integer sets. The approach is to break up the original set into a small number of subsets, find a compact polyhedral description of the convex hull of each subset, and then take the convex hull of the union of these polyhedra. The resulting extended formulation is then compact, its projection is the convex hull of the original s...

The planar Euclidean version of the traveling salesperson problem requires finding the shortest tour through a two-dimensional array of points. MacGregor and Ormerod (1996) have suggested that people solve such problems by using a global-to-local perceptual organizing process based on the convex hull of the array. We review evidence for and against this idea, before considering an alternative, ...

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let g(k) be the smallest integer such that every planar point set in general position with at least g(k) interior points has a convex subset of points with exactly k interior points of P . In this article, we prove that g(3)= 9.

We prove sharp regularity results for the convex envelope of a continuous function inside a convex domain.