Hohe Umsatzzahlen und bis zu 98% ee werden bei katalytischen Hydrierungen nichtfunktionalisierter Arylolefine mit Phosphandihydrooxazol‐Iridiumkomplexen 1 erzielt (siehe Schema). Die Anionen der Katalysatorkomplexe, z. B. Hexafluorophosphat oder Tetrakis[3,5‐bis(trifluormethyl)phenyl]borat (BARF−), haben einen starken Einfluß auf Reaktivität und Lebensdauer der Katalysatoren.

The Ham-Sandwich theorem is a well-known result in geometry. It states that any d mass distributions in R d can be simultaneously bisected by a hyperplane. The result is tight, that is, there are examples of d+1 mass distributions that cannot be simultaneously bisected by a single hyperplane. In this abstract we will study the following question: given a continuous assignment of mass distributions to certain subsets of R d , 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 type are linear subspaces of R d , i.e., k-dimensional flats containing the origin. We show that for any continuous assignment of d mass distributions to the k-dimensional linear subspaces of R d , there is always a subspace on which we can simultaneously bisect the images of all d assignments. We extend this result to center transversals, a generalization of Ham-Sandwich cuts. As for Ham-Sandwich cuts, we further show that for d − k + 2 masses, we can choose k − 1 of the vectors defining the k-dimensional subspace in which the solution lies.The second type of subsets we consider are subsets that are determined by families of n hyperplanes in R d . Also in this case, we find a Ham-Sandwich-type result. In an attempt to solve a conjecture by Langerman about bisections with several cuts, we show that our underlying topological result can be used to prove this conjecture in a relaxed setting.

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 ∈ P can see each other if the line segment seg(a, b) 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 .

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 Hamsandwich theorem: Given four mass distributions in the plane, they can be simultaneously bisected with two lines. That is, there exist two oriented lines with the following property: let R + 1 be the region of the plane that lies to the positive side of both lines and let R + 2 be the region of the plane that lies to the negative side of both lines. Then R + = R + 1 ∪ R + 2 contains exactly half of each mass distribution.

A famous result about mass partitions is the so called Ham-Sandwich theorem. It states that any d mass distributions in R d can be simultaneously bisected by a single hyperplane. In this work, we study two related questions.The first one is, whether we can bisect more than d masses, if we allow for bisections with more general objects such as cones, wedges or double wedges. We answer this question in the affirmative by showing that with all of these objects, we can simultaneously bisect d + 1 masses. For double wedges, we prove a stronger statement, namely that d families of d + 1 masses each can each by simultaneously bisected by some double wedge such that all double wedges have one hyperplane in common.The second question is, how many masses we can simultaneously equipartition with a k-fan, that is, k half-hyperplanes in R d , emanating from a common (d − 2)-dimensional apex. This question was already studied in the plane, our contribution is to extend the planar results to higher dimensions.All of our results are proved using topological methods. We use some well-established techniques, but also some newer methods. In particular, we introduce a Borsuk-Ulam theorem for flag manifolds, which we believe to be of independent interest.