For a list of my publications, visit my profiles on **Google Scholar** and **dblp**.

The
well-known Ham-Sandwich Theorem states
that any d masses in d-dimensional
space can be simultaneously bisected
by a single hyperplane. The proof of
this result uses tools from algebraic
topology, namely the famous
Borsuk-Ulam Theorem. A natural
question now is to ask, whether we can
bisect even more masses, if we use
more or different types of cuts. For
many such questions, new topological tools, similar to the
Borsuk-Ulam Theorem, are required.

Depth
measures are a way to generalize the
concept of a median to higher
dimensions. Several different depth
measures have been introduced in the
literature, all of which lead to
interesting combinatorial and
computational questions. I am
particularly interested in the
relationship between different
measures and the depth structures that
they induce on a data set in high dimensions.

Many
problems on point sets (e.g. finding a
Ham-Sandwich cut, size of the largest
family of pairwise crossing segments,...) do not depend on the
actual coordinates of the points, but
only on their relative positions, the
so-called Order Types. Among the many
open problems related with point sets,
one that caught my attention the most
is the question how many crossing-free
spanning trees can be packed into the
complete geometric graph drawn on the
given point set. This was my first
research project, started with my **Master's Thesis**

Here you find a selection of some of my recent papers.

Given a continuous assignment of mass distributions to certain subsets of d-dimensional Euclidean space, is there a subset on which we can bisect more masses than what is guaranteed by the Ham-Sandwich Theorem? We investigate two types of subsets. The first are linear subspaces, for which we show that there is always a subspace in which we can bisect as many masses as we could in the total space, proving a conjecture of Barba along the way. This result is also extended to center transversals. The second type of subsets are subsets that are defined by families of hyperplanes. Also in this case, we find a Ham-Sandwich-type result. We further use the underlying topological result to prove a conjecture by Langerman in a relaxed setting.

We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.

Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a,b in P can see each other if the line segment between then is contained in P. We denote by V(P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.