Containment problem and combinatorics

In this note, we consider two configurations of twelve lines with nineteen triple points (i.e. points where three lines meet). Both of them have the same arrangemental combinatorial features, which means that in both configurations nine of twelve lines have five triple points and one double point, and the remaining three lines have four triple points and three double points. Taking the ideal of the triple points of these configurations we discover that, quite surprisingly, for one of the configurations the containment I(3)⊂I2 holds, while for the other it does not. Hence, for ideals of points defined by arrangements of lines, the (non)containment of a symbolic power in an ordinary power is not determined alone by arrangemental combinatorial features of the configuration. Moreover, for the configuration with the non-containment I(3)⊈I2, we examine its parameter space, which turns out to be a rational curve, and thus establish the existence of a rational non-containment configuration of points. Such rational examples are very rare

On some operators acting on line arrangements and their dynamics

We study some natural operators acting on configurations of points and lines in the plane and remark that many interesting configurations are fixed points for these operators. We review ancient and recent results on line or point arrangements though the realm of these operators. We study the first dynamical properties of the iteration of these operators on some line arrangements.Comment: 33 page

Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend

Convex Optimization Techniques for Geometric Covering Problems

The present thesis is a commencement of a generalization of covering results in specific settings, such as the Euclidean space or the sphere, to arbitrary compact metric spaces. In particular we consider coverings of compact metric spaces $(X,d)$ by balls of radius $r$. We are interested in the minimum number of such balls needed to cover $X$, denoted by \Ncal(X,r). For finite $X$ this problem coincides with an instance of the combinatorial \textsc{set cover} problem, which is $\mathrm{NP}$-complete. We illustrate approximation techniques based on the moment method of Lasserre for finite graphs and generalize these techniques to compact metric spaces $X$ to obtain upper and lower bounds for \Ncal(X,r). \\ The upper bounds in this thesis follow from the application of a greedy algorithm on the space $X$. Its approximation quality is obtained by a generalization of the analysis of Chv\'atal's algorithm for the weighted case of \textsc{set cover}. We apply this greedy algorithm to the spherical case $X=S^n$ and retrieve the best non-asymptotic bound of B\"or\"oczky and Wintsche. Additionally, the algorithm can be used to determine coverings of Euclidean space with arbitrary measurable objects having non-empty interior. The quality of these coverings slightly improves a bound of Nasz\'odi. \\ For the lower bounds we develop a sequence of bounds \Ncal^t(X,r) that converge after finitely (say $\alpha\in\N$) many steps: \Ncal^1(X,r)\leq \ldots \leq \Ncal^\alpha(X,r)=\Ncal(X,r). The drawback of this sequence is that the bounds \Ncal^t(X,r) are increasingly difficult to compute, since they are the objective values of infinite-dimensional conic programs whose number of constraints and dimension of underlying cones grow accordingly to $t$. We show that these programs satisfy strong duality and derive a finite dimensional semidefinite program to approximate \Ncal^2(S^2,r) to arbitrary precision. Our results rely in part on the moment methods developed by de Laat and Vallentin for the packing problem on topological packing graphs. However, in the covering problem we have to deal with two types of constraints instead of one type as in packing problems and consequently additional work is required

The convexification effect of Minkowski summation

Let us define for a compact set $A \subset \mathbb{R}^n$ the sequence $$A(k) = \left\{\frac{a_1+\cdots +a_k}{k}: a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big).$$ It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that $A(k)$ approaches the convex hull of $A$ in the Hausdorff distance induced by the Euclidean norm as $k$ goes to $\infty$. We explore in this survey how exactly $A(k)$ approaches the convex hull of $A$, and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on $\mathbb{R}^n$, the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of $A(k)$ does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets $A$ with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence $A(k)$, and both the Hausdorff distance and effective standard deviation are eventually monotone (once $k$ exceeds $n$). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets, falsify a conjecture of Dyn and Farkhi (2004), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.Comment: 60 pages, 7 figures. v2: Title changed. v3: Added Section 7.2 resolving Dyn-Farkhi conjectur

Illuminating meaningful diversity in complex feature spaces through adaptive grid-based genetic algorithms

In many fields there exist problems for which multiple solutions of suitably high performance may be found across distinct regions of the search space. Optimisation of the search towards including these distinct solutions is important not only to understanding these spaces but also to avoiding local optima. This is the goal of a type of genetic algorithms called illumination algorithms. In Chapter 2, we demonstrate the use of an illumination algorithm in the exploration of networks sharing only a given set of structural features (valid networks). This method produces a population of valid networks that are more diverse than those produced using state of the art methods, however, it was found to be too inefficient to be usable in real-world problems. Additionally, setting an appropriate resolution of the search requires some amount of prior knowledge of the space of solutions. Addressing this problem is the focus of Chapter 3, in which we develop three extensions to the method: a) an exact method of mutation whereby only valid networks are explored, b) an adaptive mechanism for setting the resolution of the search, c) a principle for tuning mutations parameters to the search’ s resolution. We show that with these additions our method is able to increase the diversity of solutions found in significantly fewer iterations. Finally, in Chapter 4 we expand our method for use in more general problem spaces. We benchmark it against the state of the art. In all tested landscapes, we show that our method is able to identify more meaningful niches in the spaces in the same number of iterations. We conclude by highlighting the limits of our framework and discuss further directions