Every Needle Point Space Contains a Compact Convex AR-Set with no Extreme Points

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer ` ≥ 2, every sufficiently large set of points in the plane contains ` collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005].

A Compact Null Set Containing a Differentiability Point of Every Lipschitz Function

We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is constructed explicitly.

Extreme Points of the Convex Set of Joint Probability Distributions with Fixed Marginals

By using a quantum probabilistic approach we obtain a description of the extreme points of the convex set of all joint probability distributions on the product of two standard Borel spaces with fixed marginal distributions.

Extreme Points of Polar Convex Sets

A Compact Convex Set in £3 Whose Exposed Points Are of the First Category