The purpose of this paper is to present two proofs of Helly's Theorem and to use it in the proofs of several theorems classified in a group called Helly-type theorems.

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Given a closed convex set C and a point x in C, let sym(x,C) denote the symmetry value of x in C, which essentially measures how symmetric C is about the point x. Denote by sym(C) the largest value of sym(x,C) among all x in C, and let x* denote the most symmetric point in C. These symmetry measures are all invariant under linear transformation, change in inner product, etc., and so are of interest in the study of the geometry of convex sets and arise naturally in the evaluation of the complexity of interior-point methods in particular. Herein we show that when C is given by the intersection of halfspaces, i.e., C={x | Ax <= b}, then x* as well as the symmetry value of C can be computed by using linear programming. Furthermore, given an approximate analytic center of C, there is a strongly polynomial-time algorithm for approximating sym(C) to any given relative tolerance. / Singapore-MIT Alliance (SMA)

Thesis (Ph. D.)--Industrial and Systems Engineering, Georgia Institute of Technology, 2006. / Xiaoming Huo, Committee Chair ; C. F. Jeff Wu, Committee Member ; Brani Vidakovic, Committee Member ; Liang Peng, Committee Member ; Ming Yuan, Committee Member.

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Lin, Jessey. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 79-80). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.

by Zang Rui. / "Nov 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 78-82) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

The main purpose of this thesis is to find the number of convex sets on a product of two totally ordered spaces. We will give formulas to find this number for specific cases and describe a process to obtain this number for all such spaces. In the first chapter we briefly discuss the motivation behind the work presented in this thesis. Also, the definitions and notation used throughout the paper are introduced here The second chapter starts with examining the product spaces of the form {1; 2; : : : ;n} × {1; 2}. That is, we begin by analyzing a two-row by n-column space for n > N. Three separate approaches are discussed, and verified, to find the total number of convex sets on the space. A general formula is presented to obtain this total for all n. In the third chapter we take the same {1; 2; : : : ;n} × {1; 2} spaces from Chapter 2 and consider all the scenarios for adding a second disjoint convex set to the space. Adding a second convex set gives a collection of two mutually disjoint sets. Again, a general formula is presented to obtain this total number of such collections for all n. The fourth chapter takes the idea from Chapter 2 and expands it to product spaces {1; 2; : : : ;n} × {1; 2; : : : ;m} consisting of more than two rows. Here the creation of convex sets having z rows from those having z − 1 rows is exploited to obtain a model that will give the total number of z-row convex sets on any n × m space, provided the set occupies z adjacent rows. Finally, the fifth chapter describes all possible scenarios for convex sets to be placed in the {1; 2; : : : ;n}×{1; 2; : : : ;m} space. This chapter then explains the process needed to acquire a count of all convex sets on any such space as well. Chapter 5 ends by walking through this process with a concrete example, breaking it down into each scenario. We conclude by briefly summarizing the results and specifying future work we would like to further investigate, in Chapter 6.

A consequence of the Hahn-Banach theorem is the classical bipolar theorem which states that the bipolar of a subset of a locally convex vector pace equals its closed convex hull. The space $\L$ of real-valued random variables on a probability space $\OF$ equipped with the topology of convergence in measure fails to be locally convex so that - a priori - the classical bipolar theorem does not apply. In this note we show an analogue of the bipolar theorem for subsets of the positive orthant $\LO$, if we place $\LO$ in duality with itself, the scalar product now taking values in $[0, \infty]$. In this setting the order structure of $\L$ plays an important role and we obtain that the bipolar of a subset of $\LO$ equals its closed, convex and solid hull. In the course of the proof we show a decomposition lemma for convex subsets of $\LO$ into a "bounded" and "hereditarily unbounded" part, which seems interesting in its own right. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

The problem of computing the minimum distance between two convex hulls has applications to many areas including robotics, computer graphics and path planning. Moreover, determining the minimum distance between two convex hulls plays a significant role in support vector machines (SVM). In this study, a new algorithm for finding the minimum distance between two convex hulls is proposed and investigated. A convergence of the algorithm is proved and applicability of the algorithm to support vector machines is demostrated. The performance of the new algorithm is compared with the performance of one of the most popular algorithms, the sequential minimal optimization (SMO) method. The new algorithm is simple to understand, easy to implement, and can be more efficient than the SMO method for many SVM problems.