Ham-Sandwich Cuts and Center Transversals in Subspaces

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

A Center Transversal Theorem for mass assignments

In this paper, based on the ideas of Blagojevi\'c, Karasev & Magozinov, we consider an extension of the center transversal theorem to mass assignments with an improved Rado depth. In particular we substitute the marginal of a measure by a more general concept called a mass assignment over a flag manifold. Our results also allow us to solve the main problem proposed by Blagojevi\'c, Karasev & Magozinov in a linear subspace of lower dimension, as long as it is contained in a high-dimensional enough ambient space.Comment: 7 page

Computational Complexity of the ?-Ham-Sandwich Problem

?_d from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the ?-Ham-Sandwich theorem. They gave an algorithm to find the hyperplane in time O(n (log n)^{d-3}), where n is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]). Recently, Fearnley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called Unique End-of-Potential Line (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the ?-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the P-matrix linear complementarity problem

The Complexity of Sharing a Pizza

Assume you have a 2-dimensional pizza with 2n ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this fairly, that is, your friend and you should each get exactly half of each ingredient. How many cuts do you need? It was recently shown using topological methods that n cuts always suffice. In this work, we study the computational complexity of finding such n cuts. Our main result is that this problem is PPA-complete when the ingredients are represented as point sets. For this, we give a new proof that for point sets n cuts suffice, which does not use any topological methods. We further prove several hardness results as well as a higher-dimensional variant for the case where the ingredients are well-separated

Convex partitions of vector bundles and fibrewise configuration spaces

We begin this thesis with a discussion of problems from geometry and combinatorics, to which methods from equivariant algebraic topology have been successfully applied in the past. The generalised Nandakumar & Ramana~Rao problem (due to Karasev, Hubard, & Aronov and Blagojević & Ziegler) asks whether given a full-dimensional compact convex body K in R^n, n-1 continuous real functions on the space of all full-dimensional compact convex bodies in R^n and a natural number m, one can always find a partition of K into m convex pieces of equal volume such that the value of each function is equal on all the pieces of the partition. Inspired by this problem and recent parameterised generalisations of mass partition types by several groups of researchers, we formulate a parameterised version of the Nandakumar & Ramana~Rao problem, where we aim to equipart j>n-1 functions, but are allowed to choose a convex body K from some family parameterised by a vector bundle E over a CW-complex B. After we make the notion of "parameterised by the vector bundle E'' precise in Chapter~II, we follow the strategy developed by Karasev, Hubard & Aronov to formulate a topological criterion for the existence of solutions to the parameterised Nandakumar & Ramana Rao problem. Due to the limitations of our topological methods, we restrict our attention to the case when m equals some prime p. Chapter~III contains a brief overview of various standard algebraic topology results that we use extensively in the later chapters. In Chapter~IV we extend the results of Jaworowski concerning Fadell-Husseini indices of sphere bundles, equipped with free fibrewise action of the cyclic group Z/Z_p, by considering the symmetric group S_p. Next, we compute the index of the fibrewise configuration space Fconf(p,E) of p distinct points with respect to S_p in the case of vector bundle E of an odd rank. In the case when E has an even rank, we provide bounds on the index, showing that the upper bound is tight in some cases. Then we change the group that acts on E, and compute the index of the space Fconf(p, E) with respect to cyclic group action in the special case when E admits two linearly independent nowhere zero sections. In Chapter~V we use these computations to find a partial solution to the parameterised Nandakumar & Ramana~Rao problem. For any pair of a vector bundle E and a prime p, we describe a range of j such that the parameterised Nandakumar & Ramana~Rao has a solution for the family of convex bodies parameterised by E, the desired number p of pieces in partition and a choice of j appropriately defined continuous functions. Finally, we apply these computations to the case of tautological bundles over the Grassmannians.Die Dissertation beginnt mit einer Diskussion einiger geometrischer und kombinatorischer Probleme, zu deren Studium sich Methoden der Äquivarianten Algebraischen Topologie in der Vergangenheit als geeignet erwiesen haben. Das verallgemeinerte Nandakumar & Ramana~Rao-Problem (nach Karasev, Hubard & Aronov, sowie Blagojević & Ziegler) besteht in der Frage, ob sich für einen gegebenen volldimensionalen, kompakten, konvexen Körper K im R^n, eine Familie von n-1 stetigen reellen Funktionen auf dem Raum aller solchen Körper im R^n, sowie eine gegebene natürliche Zahl m stets eine Partition von K in m konvexe Teilmengen gleichen Volumens finden lässt, derart, dass jede einzelne der Funktionen auf allen Teilen gleiche Werte annimmt. Wir formulieren eine parametrisierte Version dieses Problems, die nach einer Gleichteilung bezüglich einer Familie von möglicherweise mehr als n-1 Funktionen fragt, allerdings erlaubt, den konvexen Körper K aus einer durch ein Vektorbündel E über einem CW-Komplex parametrisierten Familie zu wählen. Nachdem wir im zweiten Kapitel den Begriff der “Parametrisierung durch ein Vektorbündel” präzisieren, verfolgen wir die von Karasev, Hubard & Aronov entwickelte Strategie, topologische Kriterien für die Lösbarkeit des parametriserten Nandakumar & Ramana~Rao-Problems in gewissen Fällen zu finden. Den Grenzen unserer topologischen Methoden ist es geschuldet, dass wir uns dabei auf den Fall beschränken, indem m=p eine Primzahl ist. Das dritte Kapitel der Arbeit gibt einen Überblick über verschiedene bekannte Ergebnisse der Algebraischen Topologie, die wir in den späteren Kapiteln benutzen werden. Im vierten Kapitel erweitern wir Ergebnisse Jaworowskis über Fadell-Husseini-Indizes gewisser Sphärenbündeln, die mit einer faserweisen Wirkung der zyklischen Gruppe Z/Z_p ausgestattet sind, wobei eine Wirkung der symmetrischen Gruppe S_p an die Stelle der Z/Z_p-Wirkung tritt. Als Nächstes berechnen wir die Indizes (bzgl. S_p-Wirkung) des Faserweisen Konfigurationsraums Fconf(p, E) von p Punkten in einem Vektorbündel E ungeraden Rangs, geben für den Fall geraden Rangs (teils bestmögliche) Schranken an, und berechnen den Index (bzgl. Z/Z_p oder S_p-Wirkung) im Spezialfall, dass E zwei linear unabhängige Schnitte zulässt. Das fünfte Kapitel behandelt, wie die Ergebnisse unserer Berechnungen zu einer teilweisen Lösung des parametrisierten Nandakumar & Ramana~Rao-Problems führen. Für jedes Paar, bestehend aus einem Vektorbündel E und einer Primzahl p beschreiben wir einen Bereich möglicher Werte von j, für die das parametrisierte Nandakumar & Ramana~Rao-Problem bezüglich dem Tripel (E, p, j) eine Lösung besitzt. Schließlich wenden wir unsere Überlegungen auf den Spezialfall Tautologischer Bündel über Grassmann-Mannigfaltigkeiten an