Longest k-monotone chains

We study higher order convexity properties of random point sets in the unit square. Given $n$ uniform i.i.d random points, we derive asymptotic estimates for the maximal number of them which are in $k$-monotone position, subject to mild boundary conditions. Besides determining the order of magnitude of the expectation, we also prove strong concentration estimates. We provide a general framework that includes the previously studied cases of $k=1$ (longest increasing sequences) and $k=2$ (longest convex chains)

Analytic and probabilistic problems in discrete geometry

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence u1,...,un of norm 1 vectors in a real Hilbert space H , there exists a unit vector \vartheta \epsilon H , such that \sum 1 over [ui, v]2 \leqslant n2. The 2-dimensional case is proved by complex analytic methods. For the higher dimensional extremal cases, we prove a tensorisation result that is similar to F. John's theorem about characterisation of ellipsoids of maximal volume. From this, we deduce that the only full dimensional locally extremal system is the orthonormal system. We also obtain the same result for the weaker, original polarization problem. The second chapter investigates a problem in probabilistic geometry. Take n independent, uniform random points in a triangle T. Convex chains between two fixed vertices of T are defined naturally. Let Ln denote the maximal size of a convex chain. We prove that the expectation of Ln is asymptotically \alpha n1/3, where \alpha is a constant between 1:5 and 3:5 - we conjecture that the correct value is 3. We also prove strong concentration results for Ln, which, in turn, imply a limit shape result for the longest convex chains

Active Learning a Convex Body in Low Dimensions

Consider a set $P \subseteq \Re^d$ of $n$ points, and a convex body $C$ provided via a separation oracle. The task at hand is to decide for each point of $P$ if it is in $C$ using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using $O( h(P) \log n)$ queries, where $h(P)$ is the largest subset of points of $P$ in convex position. Furthermore, we show that in two dimensions one can solve this problem using $O( v(P,C) \log^2 n )$ oracle queries, where $v(P, C)$ is a lower bound on the minimum number of queries that any algorithm for this specific instance requires.Comment: Talk based on results in the paper is available here: https://youtu.be/5Epyh2lHrF

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

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page