Convex Hulls, Oracles, and Homology

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the ``no''-case of POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).Comment: 11 pages, 2 figure

Об одном подходе к построению выпуклой оболочки конечного множества точек в Rⁿ

В статье предложен метод построения выпуклой оболочки конечного множества точек в Rⁿ , позволяющий решать задачи, не требующие описания всех подграней границы выпуклой оболочки. Описаны основные процедуры построения выпуклой оболочки, представленной в виде n-политопа, заданного пересечением замкнутых полупространств. Приведены численные результаты работы метода при n = 4; 5.У статті запропоновано метод побудови опуклої оболонки кінцевої множини точок в Rⁿ , що дозволяє вирішувати завдання, які не вимагають опису всіх підграней границі опуклої оболонки. Описано основні процедури побудови опуклої оболонки, представленої у вигляді n-політопа, що заданий перетином замкнутих півпросторів. Наведено чисельні результати роботи методу при n = 4; 5.In article the method of construction of a convex hull of points finite set in Rⁿ , allowing is offered to solve problems not requiring descriptions all subfaces of border of a convex hull. The basic procedures of construction of a convex hull submitted as n-polytope, given by crossing closed half-spaces are described. The numerical results of operation of a method at n = 4; 5 are received

Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For inputs in general position the number of bounded faces is O(n^d). For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a polynomial number of linear programs