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.
A centerpoint is a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for a point set. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median. We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set of (few) points such that every halfspace that contains one point of them contains a large fraction of the data points and every halfspace that contains more of them contains an even larger fraction of the data points.
Assume you have a pizza consisting of four ingredients (e.g., bread, tomatoes, cheese and olives) that you want to share with your friend. You want to do this fairly, meaning that you and your friend should get the same amount of each ingredient. How many times do you need to cut the pizza so that this is possible? We will show that two straight cuts always suffice. More formally, we will show the following extension of the well-known Ham-sandwich theorem: Given four mass distributions in the plane, they can be simultaneously bisected with two lines.