Barany I.,MTA Renyi Institute of Mathematics |
Barany I.,University College London |
Roldan-Pensado E.,MTA Renyi Institute of Mathematics |
Roldan-Pensado E.,National Autonomous University of Mexico |
Toth G.,MTA Renyi Institute of Mathematics
Discrete and Computational Geometry | Year: 2015
According to the Erdős–Szekeres theorem, for every n, a sufficiently large set of points in general position in the plane contains n in convex position. In this note we investigate the line version of this result, that is, we want to find n lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erdős–Szekeres theorem. © 2015, Springer Science+Business Media New York.
Csiszar I.,MTA Renyi Institute of Mathematics |
Breuer T.,General Electric
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2015
Familiar approaches to risk and preferences involve minimizing the expectation EIP(X) of a payoff function X over a family Γ of plausible risk factor distributions IP. We consider Γ determined by a bound on a convex integral functional of the density of IP, thus Γ may be an I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution IP0. Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing EIP(X) subject to IP ∈ Γ), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When Γ is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls. © Springer International Publishing Switzerland 2015.