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In this paper, we study a problem of mass partitions by parallel hyperplanes. Takahashi and Sober\'on conjectured an extension of the classical ham sandwich theorem: any $d+k-1$ measures in $\mathbb{R}^d$ can be simultaneously equipartitioned by $k$ parallel hyperplanes. We construct a configuration space -- test map scheme and prove a new Borsuk-Ulam-type theorem to show that the conjecture is true in the case when the Stirling number of second kind $S(d+k-1, k)$ is odd. This recovers exactly the parity condition obtained by Hubard and Sober\'on via different methods, reinforcing the possibility that this condition is both necessary and sufficient. Our proof relies on a novel computation of the Fadell-Husseini index., Comment: 14 pages, 3 figures

The Tverberg--Vre\'cica conjecture is a broad generalization of Tverberg's classical theorem. One of its consequences, the central transversal theorem, extends both the centerpoint theorem and the ham sandwich theorem. In this manuscript, we establish complex analogues of these results where the corresponding transversals are complex affine spaces. The proof of the complex Tverberg--Vre\'cica conjecture for powers of the same prime relies on the non-vanishing of an equivariant Euler class. Additionally, we obtain new Borsuk--Ulam-type theorems on complex Stiefel manifolds, which are interesting on their own. These theorems yield complex analogues of recent extensions of the ham sandwich theorem for mass assignments by Axelrod-Freed and Sober\'on, and provide a direct proof of the complex central transversal theorem., Comment: 19 pages

We show how, using linear-algebraic tools developed to prove Tverberg's theorem in combinatorial geometry, we can design new models of multi-class support vector machines (SVMs). These supervised learning protocols require fewer conditions to classify sets of points, and can be computed using existing binary SVM algorithms in higher-dimensional spaces, including soft-margin SVM algorithms. We describe how the theoretical guarantees of standard support vector machines transfer to these new classes of multi-class support vector machines. We give a new simple proof of a geometric characterization of support vectors for largest margin SVMs by Veelaert., Comment: 15 pages, 3 figures

We prove a common generalization to several mass partition results using hyperplane arrangements to split $\mathbb{R}^d$ into two sets. Our main result implies the ham-sandwich theorem, the necklace splitting theorem for two thieves, a theorem about chessboard splittings with hyperplanes with fixed directions, and all known cases of Langerman's conjecture about equipartitions with $n$ hyperplanes. Our main result also confirms an infinite number of previously unknown cases of the following conjecture of Takahashi and Sober\'on: For any $d+k-1$ measures in $\mathbb{R}^d$, there exist an arrangement of $k$ parallel hyperplanes that bisects each of the measures. The general result follows from the case of measures that are supported on a finite set with an odd number of points. The proof for this case is inspired by ideas of differential and algebraic topology, but it is a completely elementary parity argument., Comment: 18 pages, 3 figures

We prove extensions of Halman's discrete Helly theorem for axis-parallel boxes in $\mathbb{R}^d$. Halman's theorem says that, given a set $S$ in $\mathbb{R}^d$, if $F$ is a finite family of axis-parallel boxes such that the intersection of any $2d$ contains a point of $S$, then the intersection of $F$ contains a point of $S$. We prove colorful, fractional, and quantitative versions of Halman's theorem. For the fractional versions, it is enough to check that many $(d+1)$-tuples of the family contain points of $S$. Among the colorful versions we include variants where the coloring condition is replaced by an arbitrary matroid. Our results generalize beyond axis-parallel boxes to $H$-convex sets., Comment: 13 pages, 1 figure

A theorem of Gr\"unbaum, which states that every $m$-polytope is a refinement of an $m$-simplex, implies the following generalization of Tverberg's theorem: if $f$ is a linear function from an $m$-dimensional polytope $P$ to $\mathbb{R}^d$ and $m \ge (d + 1)(r - 1)$, then there are $r$ pairwise disjoint faces of $P$ whose images intersect. Moreover, the topological Tverberg theorem implies that this statement is true whenever the map $f$ is continuous and $r$ is a prime power. In this note, we show that for certain families of polytopes the lower bound on the dimension $m$ of the polytopes can be significantly improved, both in the affine and topological cases., Comment: 10 pages

A vast array of envy-free results have been found for the subdivision of one-dimensional resources, such as the interval $[0,1]$. The goal is to divide the space into $n$ pieces and distribute them among $n$ observers such that each receives their favorite pieces. We study high-dimensional versions of these results. We prove that several spaces of convex partitions of $\mathbb{R}^d$ allow for envy-free division among any $n$ observers. We also prove the existence of convex partitions of $\mathbb{R}^d$ which allow for envy-free divisions among several groups of $n$ observers simultaneously., Comment: 14 pages, 3 figures

Given a finite set of points in $\mathbb{R}^d$, Tverberg's theorem guarantees the existence of partitions of this set into parts whose convex hulls intersect. We introduce a graph structured on the family of Tverberg partitions of a given set of points, whose edges describe closeness between different Tverberg partitions. We prove bounds on the minimum and maximum degree of this graph, the number of vertices of maximal degree, its clique number, and its connectedness.

Krasnosselsky's art gallery theorem gives a combinatorial characterization of star-shaped sets in Euclidean spaces, similar to Helly's characterization of finite families of convex sets with non-empty intersection. We study colorful and quantitative variations of Krasnosselsky's result. In particular, we are interested in conditions on a set $K$ that guarantee there exists a measurably large set $K'$ such that every point in $K'$ can see every point in $K$. We prove results guaranteeing the existence of $K'$ with large volume or large diameter., Comment: 12 pages, 4 Figures

Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion that measures how good of a regression hyperplane a given query hyperplane is with respect to a set of data points. Under projective duality, this can be interpreted as a depth measure for query points with respect to an arrangement of data hyperplanes. The study of depth measures for query points with respect to a set of data points has a long history, and many such depth measures have natural counterparts in the setting of hyperplane arrangements. For example, regression depth is the counterpart of Tukey depth. Motivated by this, we study general families of depth measures for hyperplane arrangements and show that all of them must have a deep point. Along the way we prove a Tverberg-type theorem for hyperplane arrangements, giving a positive answer to a conjecture by Rousseeuw and Hubert from 1999. We also get three new proofs of the centerpoint theorem for regression depth, all of which are either stronger or more general than the original proof by Amenta, Bern, Eppstein, and Teng. Finally, we prove a version of the center transversal theorem for regression depth., Comment: To be presented at the 39th International Symposium on Computational Geometry (SoCG 2023)

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., Comment: 10 pages, 2 figures

We use recent extensions of the Borsuk--Ulam theorem for Stiefel manifolds to generalize the ham sandwich theorem to mass assignments. A $k$-dimensional mass assignment continuously imposes a measure on each $k$-dimensional affine subspace of $\mathbb{R}^d$. Given a finite collection of mass assignments of different dimensions, one may ask if there is some sequence of affine subspaces $S_{k-1} \subset S_k \subset \ldots \subset S_{d-1} \subset \mathbb{R}^d$ such that $S_i$ bisects all the mass assignments on $S_{i+1}$ for every $i$. We show it is possible to do so whenever the number of mass assignments of dimensions $(k,\ldots,d)$ is a permutation of $(k,\ldots,d)$. We extend previous work on mass assignments and the central transversal theorem. We also study the problem of halving several families of $(d-k)$-dimensional affine spaces of $\mathbb{R}^d$ using a $(k-1)$-dimensional affine subspace contained in some translate of a fixed $k$-dimensional affine space. For $k=d-1$, there results can be interpreted as dynamic ham sandwich theorems for families of moving points., Comment: 17 pages, 7 figures

Many results in mass partitions are proved by lifting $\mathbb{R}^d$ 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 $\mathbb{R}^d$ by parallel hyperplanes and of $d+2$ measures in $\mathbb{R}^d$ 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., Comment: 16 pages, 3 figures

In 2008 Karasev conjectured that for every set of $r$ blue lines, $r$ green lines, and $r$ red lines in the plane, there exists a partition of them into $r$ colorful triples whose induced triangles intersect. We disprove this conjecture for every $r$ and extend the counterexamples to higher dimensions., Comment: 5 pages, 1 figure

We generalize the Yao-Yao partition theorem by showing that for any smooth measure in $R^d$ there exist equipartitions using $(t+1)2^{d-1}$ convex regions such that every hyperplane misses the interior of at least $t$ regions. In addition, we present tight bounds on the smallest number of hyperplanes whose union contains the boundary of an equipartition of a measure into $n$ regions. We also present a simple proof of a Borsuk-Ulam type theorem for Stiefel manifolds that allows us to generalize the central transversal theorem and prove results bridging the Yao--Yao partition theorem and the central transversal theorem., Comment: 18 pages, 3 figures

For a finite set $S$ of points in the plane and a graph with vertices on $S$ consider the disks with diameters induced by the edges. We show that for any odd set $S$ there exists a Hamiltonian cycle for which these disks share a point, and for an even set $S$ there exists a Hamiltonian path with the same property. We discuss high-dimensional versions of these theorems and their relation to other results in discrete geometry., Comment: 8 pages, 3 figures

We show that for any compact convex set $K$ in $\mathbb{R}^d$ and any finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$, if the intersection of every sufficiently small subfamily of $\mathcal{F}$ contains an isometric copy of $K$ of volume $1$, then the intersection of the whole family contains an isometric copy of $K$ scaled by a factor of $(1-\varepsilon)$, where $\varepsilon$ is positive and fixed in advance. Unless $K$ is very similar to a disk, the shrinking factor is unavoidable. We prove similar results for affine copies of $K$. We show how our results imply the existence of randomized algorithms that approximate the largest copy of $K$ that fits inside a given polytope $P$ whose expected runtime is linear on the number of facets of $P$., Comment: 10 pages, 2 figures

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., Comment: 37 pages, 8 figures, 1 table

A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a longstanding 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., Comment: 11 pages, 1 figure

Tverberg's theorem bounds the number of points $\mathbb{R}^d$ needed for the existence of a partition into $r$ parts whose convex hulls intersect. If the points are colored with $N$ colors, we seek partitions where each part has at most one point of each color. In this manuscript, we bound the number of color classes needed for the existence of partitions where the convex hulls of the parts intersect even after any set of $t$ colors is removed. We prove asymptotically optimal bounds for $t$ when $r \le d+1$, improve known bounds when $r>d+1$, and give a geometric characterization for the configurations of points for which $t=N-o(N)$., Comment: 13 pages, 1 figure

Tverberg-type theory aims to establish sufficient conditions for a simplicial complex $\Sigma$ such that every continuous map $f\colon \Sigma \to \mathbb{R}^d$ maps $q$ points from pairwise disjoint faces to the same point in $\mathbb{R}^d$. Such results are plentiful for $q$ a power of a prime. However, for $q$ with at least two distinct prime divisors, results that guarantee the existence of $q$-fold points of coincidence are non-existent -- aside from immediate corollaries of the prime power case. Here we present a general method that yields such results beyond the case of prime powers. In particular, we prove previously conjectured upper bounds for the topological Tverberg problem for all $q$., Comment: The proof of Theorem 1.1 is incomplete. We are grateful to an anonymous referee for pointing out a mistake in our proof. This revised version briefly explains the gap in the proof and otherwise reproduces the original version

We prove an extension of a ham sandwich theorem for families of lines in the plane by Dujmovi\'{c} and Langerman. Given two sets $A, B$ of $n$ lines each in the plane, we prove that it is possible to partition the plane into $r$ convex regions such that the following holds. For each region $C$ of the partition there is a subset of $c_r n^{1/r}$ lines of $A$ whose pairwise intersections are in $C$, and the same holds for $B$. In this statement $c_r$ only depends on $r$. We also prove that the dependence on $n$ is optimal., Comment: 14 pages, 4 figures

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colorful and fractional variants of all our Helly-type theorems., Comment: 25 pages, 2 figures

We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope., Comment: 12 pages, 1 figure

We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When $\Gamma$ is a subgroup of the combinatorial automorphism group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with combinatorial automorphism group exactly $\Gamma$. When $\Gamma$ is a subgroup of the geometric symmetry group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with both geometric symmetry group and combinatorial automorphism group exactly $\Gamma$. These symmetry-breaking results then are applied to show that for every abelian group $\Gamma$ of even order and every involution $\sigma$ of $\Gamma$, there is a centrally symmetric convex polytope with geometric symmetry group $\Gamma$ such that $\sigma$ corresponds to the central symmetry., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1505.06253

The maximum rectilinear crossing number of a graph $G$ is the maximum number of crossings in a good straight-line drawing of $G$ in the plane. In a good drawing any two edges intersect in at most one point (counting endpoints), no three edges have an interior point in common, and edges do not contain vertices in their interior. A spider is a subdivision of $K_{1,k}$. We provide both upper and lower bounds for the maximum rectilinear crossing number of spiders. While there are not many results on the maximum rectilinear crossing numbers of infinite families of graphs, our methods can be used to find the exact maximum rectilinear crossing number of $K_{1,k}$ where each edge is subdivided exactly once. This is a first step towards calculating the maximum rectilinear crossing number of arbitrary trees., Comment: 8 pages

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., Comment: 48 pages, 8 figures

We prove a Tverberg-type theorem using the probabilistic method. Given $\varepsilon >0$, we find the smallest number of partitions of a set $X$ in $R^d$ into $r$ parts needed in order to induce at least one Tverberg partition on every subset of $X$ with at least $\varepsilon |X|$ elements. This generalizes known results about Tverberg's theorem with tolerance., Comment: 14 pages

Holmsen, Kyn\v{c}l and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint. We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geq 2$, $d\geq 2$, $c\geq d$ with $m\geq n(c-d)+d$ are integers, and $\mu_1, \dots, \mu_m$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^d$, we prove that there exists a partition of $\mathbb{R}^d$ into $n$ convex pieces which equiparts the measures $\mu_1, \dots, \mu_{d-1}$, and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $\mu_1, \dots, \mu_m$., Comment: 20 pages, 5 figures; new coauthor; revised extended version with stronger results

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 $mr/d$ 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., Comment: 14 pages, 9 figures

We prove a Tverberg type theorem: Given a set $A \subset \mathbb{R}^d$ in general position with $|A|=(r-1)(d+1)+1$ and $k\in \{0,1,\ldots,r-1\}$, there is a partition of $A$ into $r$ sets $A_1,\ldots,A_r$ with the following property. The unique $z \in \bigcap_1^r \mathrm{aff} A_j$ can be written as an affine combination of the element in $A_j$: $z = \sum_{x \in A_j} \alpha(x)x$ for every $p$ and exactly $k$ of the coefficients $\alpha(x)$ are negative. The case $k=0$ is Tverberg's classical theorem.

We use the probabilistic method to obtain versions of the colorful Carath\'eodory 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 $R^d$, 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 the bound $N=rt+O(\sqrt{t})$ for fixed $r,d$ which is polynomial in each parameters. Our bounds extend to colorful versions of Tverberg's theorem, as well as Reay-type variations of this theorem., Comment: 18 pages

We obtain algorithms for computing Tverberg partitions based on centerpoint approximations. This applies to a wide range of convexity spaces, from the classic Euclidean setting to geodetic convexity in graphs. In the Euclidean setting, we present probabilistic algorithms which are weakly polynomial in the number of points and the dimension. For geodetic convexity in graphs, we obtain deterministic algorithms for cactus graphs and show that the general problem of finding the Radon number is NP-hard., Comment: 17 pages

We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional and $(p,q)$ versions work with conditions where the corresponding Helly theorem does not. We also include variants of Tverberg's theorem, B\'ar\'any's point selection theorem and the existence of weak epsilon-nets for convex sets with diameter estimates., Comment: 16 pages

This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization., Comment: 40 pages, 1 figures

Given a finite set $X$ of points in $R^n$ and a family $F$ of sets generated by the pairs of points of $X$, we determine volumetric and structural conditions for the sets that allow us to guarantee the existence of a positive-fraction subfamily $F'$ of $F$ for which the sets have non-empty intersection. This allows us to show the existence of weak epsilon-nets for these families. We also prove a topological variation of the existence of weak epsilon-nets for convex sets., Comment: 15 pages

We show quantitative versions of classic results in discrete geometry, where the size of a convex set is determined by some non-negative function. We give versions of this kind for the selection theorem of B\'ar\'any, the existence of weak epsilon-nets for convex sets and the $(p,q)$ theorem of Alon and Kleitman. These methods can be applied to functions such as the volume, surface area or number of points of a discrete set. We also give general quantitative versions of the colorful Helly theorem for continuous functions., Comment: 24 pages

We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain. As a corollary, we obtain quasi-polynomial algorithms for computing partition functions for independent sets, perfect matchings, Hamiltonian cycles and dense subgraphs in graphs as well as for graph colorings. This allows us to tell apart in quasi-polynomial time graphs that are sufficiently far from having a structure of a given type (i.e., independent set of a given size, Hamiltonian cycle, etc.) from graphs that have sufficiently many structures of that type, even when the probability to hit such a structure at random is exponentially small., Comment: constants are improved, other minor improvements

We study nested partitions of $R^d$ obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing the parts among $k$ sets. This generalises classical necklace splitting results and their more recent high-dimensional versions. With similar methods we show that in the plane, for any $t$ measures there is a path formed only by horizontal and vertical segments using at most $t-1$ turns that splits them by half simultaneously, and optimal mass-partitioning results for chessboard-colourings of $R^d$ using hyperplanes with fixed directions.

In this paper, we study the number of compact sets needed in an infinite family of convex sets with a local intersection structure to imply a bound on its piercing number, answering a conjecture of Erd\H{o}s and Gr\"unbaum. Namely, if in an infinite family of convex sets in $\mathbb{R}^d$ we know that out of every $p$ there are $q$ which are intersecting, we determine if having some compact sets implies a bound on the number of points needed to intersect the whole family. We also study variations of this problem., Comment: 11 pages, 2 figures