Related papers: Convex Hulls of Algebraic Sets

This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the…

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…

In this article, a new solution for the convex hull problem has been presented. The convex hull is a widely known problem in computational geometry. As nature is a rich source of ideas in the field of algorithms, the solution has been…

We study a natural extension to the well-known convex hull problem by introducing multiplicity: if we are given a set of convex polygons, and we are allowed to partition the set into multiple components and take the convex hull of each…

We consider polyhedral approximations of strictly convex compacta in finite dimensional Euclidean spaces (such compacta are also uniformly convex). We obtain the best possible estimates for errors of considered approximations in the…

The simplest way to generate a lattice of convex sets is to consider an initial set of points and draw segments, triangles, and any convex hull from it, then intersect them to obtain new points, and so forth. The result is an infinite…

This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…

In this paper we present several results on the expected complexity of a convex hull of $n$ points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of $n$ points,…

We consider the nonconvex set $\mathcal S_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \geq 0,\; z \in \{0,1\}^n\}$, which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best…

In this paper, an effective method with time complexity of $\mathcal{O}(K^{3/2}N^2\log \frac{K}{\epsilon_0})$ is introduced to find an approximation of the convex hull for $N$ points in dimension $n$, where $K$ is close to the number of…

Convexity prior is one of the main cue for human vision and shape completion with important applications in image processing, computer vision. This paper focuses on characterization methods for convex objects and applications in image…

Let $P$ be a set of $n$ points in $\mathbb{R}^3$ in general position, and let $RCH(P)$ be the rectilinear convex hull of $P$. In this paper we obtain an optimal $O(n\log n)$-time and $O(n)$-space algorithm to compute $RCH(P)$. We also…

We compute the volumes of convex bodies that are given by inequalities of concave polynomials. These volumes are found to arbitrary precision thanks to the representation of periods by linear differential equations. Our approach rests on…

Region extraction is necessary in a wide range of applications, from object detection in autonomous driving to analysis of subcellular morphology in cell biology. There exist two main approaches: convex hull extraction, for which exact and…

We present a numerical method for the solution of Newton's problem of least resistance in the class of convex functions using a convex hull approach. We observe that the numerically computed solutions possess some symmetry. Further, their…

The convex hull peeling of a point set consists in taking the convex hull, then removing the extreme points and iterating that procedure until no point remains. The boundary of each hull is called a layer. Following on from [15], we study…

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed…

Kippenhahn discovered that the numerical range of a complex square matrix is the convex hull of a plane real algebraic curve. Here, we present an example of a convex set, which has a similar algebraic description as the numerical range,…

The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes. Using the polymake system we explore various algorithms and…

With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the convex hull of the set of binary points satisfying a collection of equations containing…