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Many results about mass partitions are proved by lifting |$\mathds {R}^d$| to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other results, we prove the existence of equipartitions of |$d+1$| measures in |$\mathds {R}^d$| by parallel hyperplanes and of |$d+2$| measures in |$\mathds {R}^d$| by concentric spheres. For measures whose supports are sufficiently well separated, we prove results where one can cut a fixed (possibly different) fraction of each measure either by parallel hyperplanes, concentric spheres, convex polyhedral surfaces of few facets, or convex polytopes with few vertices. [ABSTRACT FROM AUTHOR]

We study the existence of fair distributions when we have more guests than pieces to allocate, focusing on envy-free distributions among those who receive a piece. The conditions on the demand from the guests can be weakened from those of classic cake-cutting and rent-splitting results of Stromquist, Woodall, and Su. We extend existing variations of the cake-cutting problem with secretive guests and those that resist the removal of any sufficiently small set of guests. [ABSTRACT FROM AUTHOR]

Mass partition problems describe the partitions we can induce on a family of measures or finite sets of points in Euclidean spaces by dividing the ambient space into pieces. In this survey we describe recent progress in the area in addition to its connections to topology, discrete geometry, and computer science. [ABSTRACT FROM AUTHOR]

A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in Rd given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a long-standing conjecture by B\'ar\'any, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of "containing k colinear integer points. [ABSTRACT FROM AUTHOR]

This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. [ABSTRACT FROM AUTHOR]

We use the probabilistic method to obtain versions of the colourful Carathéodory theorem and Tverberg's theorem with tolerance. In particular, we give bounds for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect. We prove a bound N = rt + O(√t) for fixed r,d which is polynomial in each parameters. Our bounds extend to colourful versions of Tverberg's theorem, as well as Reay-type variations of this theorem. [ABSTRACT FROM AUTHOR]

Abstract: We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of m measures in R d among r thieves using roughly m r / d convex pieces, even in the cases when m is larger than the dimension. The main proof relies on a construction of a geometric realization of the topological join of two spaces of partitions of R d into convex parts, and the computation of the Fadell–Husseini ideal valued index of the resulting spaces. [ABSTRACT FROM AUTHOR]

This paper presents a new variation of Tverberg's theorem. Given a discrete set S of $$\mathbb {R}^{d}$$ , we study the number of points of S needed to guarantee the existence of an m-partition of the points such that the intersection of the m convex hulls of the parts contains at least k points of S. The proofs of the main results require new quantitative versions of Helly's and Carathéodory's theorems. [ABSTRACT FROM AUTHOR]

We prove variations of Carathéodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász's colorful Helly's theorem, Bárány's colorful Carathéodory's theorem, and the colorful Tverberg's theorem. [ABSTRACT FROM AUTHOR]

We prove the following generalization of the ham sandwich theorem, conjectured by Imre Bárány. Given a positive integer k and d nice measures μ1,μ2,…,μd in ℝd such that μi(ℝd)=k for all i, there is a partition of ℝd into k interior-disjoint convex parts C1,C2,…,Ck such that μi (Cj)=1 for all i,j. If k=2 , this gives the ham sandwich theorem. This result was proved independently by R. N. Karasev. [ABSTRACT FROM PUBLISHER]