Andreas F. Holmsen

Points surrounding the origin

Suppose d > 2, n > d+1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any p ∈ P, the convex hull of Q∪{p} does not contain the origin in its interior.We also show that for non-empty, finite point sets A1, ..., Ad+1 in ℝd, if the origin is contained in the convex hull of Ai ∪ Aj for all 1≤i<j≤d+1, then there is a simplex S containing the origin such that |S∩Ai|=1 for every 1≤i≤d+1. This is a generalization of Bárány’s colored Carathéodory theorem, and in a dual version, it gives a spherical version of Lovász’ colored Helly theorem.

Intersecting convex sets by rays

What is the smallest number τ = τ(<i>n</i>) such that for any collection of <i>n</i> pairwise disjoint convex sets in <i>d</i>-dimensional Euclidean space, there is a point such that any ray (half-line) emanating from it meets at most τ sets of the collection? This question of Urrutia is closely related to the notion of regression depth introduced by Rousseeuw and Hubert (1996). We show the following: Given any collection <i>C</i> of <i>n</i> pairwise disjoint compact convex sets in <i>d</i>-dimensional Euclidean space, there exists a point <i>p</i> such that any ray emanating from <i>p</i> meets at most <i>dn</i>+1)/<i>d</i>+1) members of <i>C</i>. There exist collections of n pairwise disjoint (i) equal length segments or (ii) disks in the Euclidean plane such that from any point there is a ray that meets at least 2<i>n</i>/3--2 of them. We also determine the asymptotic behavior of τ(<i>n</i>) when the convex bodies are fat and of roughly equal size.

THE ERDŐS–SZEKERES PROBLEM FOR NON-CROSSING CONVEX SETS

We show an equivalence between a conjecture of Bisztriczky and Fejes Toth about families of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and Toth on the Erdős–Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk on the Erdős–Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erdős–Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erdős–Szekeres theorem of Por and Valtr to families of non-crossing convex bodies.

A geometric Hall-type theorem

We introduce a geometric generalization of Halls marriage theorem. For any family

Topology of Geometric Joins

F = \{X_1, \dots, X_m\}

Geometric Transversal Theory: T (3)-Families in the Plane

of finite sets in

New results for T (k)-families in the plane

\mathbb{R}^d

Near equipartitions of colored point sets

, we give conditions under which it is possible to choose a point

ORTHOGONAL COLORINGS OF THE SPHERE

x_i\in X_i

Cutting convex curves

for every