No-dimension Tverberg's theorem and its corollaries in Banach spaces of type p

We continue our study of ‘no‐dimension’ analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper (Adiprasito, Bárány and Mustafa, ‘Theorems of Carathéodory, Helly, and Tverberg without dimension’, Proceedings of the Thirtieth Annual ACM‐SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, San Diego, California, 2019) 2350–2360) and prove no‐dimension versions of the colored Tverberg theorem, the selection lemma and the weak ‐net theorem in Banach spaces of type >1 . To prove these results, we use the original ideas of Adiprasito, Bárány and Mustafa for the Euclidean case, our no‐dimension version of the Radon theorem and slightly modified version of the celebrated Maurey lemma

Theorems of Carathéodory, Helly, and Tverberg without dimension

We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point a∈convP, and an integer r≤n, there is a subset Q⊂P of r elements such that the distance between a and convQ is less than diamP/2r−−√. In an analoguos no-dimension Helly theorem a finite family F of convex bodies is given, all of them are contained in the Euclidean unit ball of Rd. If k≤d, |F|≥k, and every k-element subfamily of F is intersecting, then there is a point q∈Rd which is closer than 1/k−−√ to every set in F. This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established

No-Dimensional Tverberg Theorems and Algorithms

Tverberg's theorem is a classic result in discrete geometry. It states that for any integer $k \ge 2$ and any finite $d$-dimensional point set $P \subseteq \mathbb{R}^d$ of at least $(d + 1)(k - 1) + 1$ points, we can partition $P$ into $k$ subsets whose convex hulls have a non-empty intersection. The computational problem of finding such a partition lies in the complexity class $\mathrm{PPAD} \cap \mathrm{PLS}$, but no hardness results are known. Tverberg's theorem also has a colorful variant: the points in $P$ have colors, and under certain conditions, $P$ can be partitioned into colorful sets, i.e., sets in which each color appears exactly once such that the convex hulls of the sets intersect. Recently, Adiprasito, Barany, and Mustafa [SODA 2019] proved a no-dimensional version of Tverberg's theorem, in which the convex hulls of the sets in the partition may intersect in an approximate fashion, relaxing the requirement on the cardinality of $P$. The argument is constructive, but it does not result in a polynomial-time algorithm. We present an alternative proof for a no-dimensional Tverberg theorem that leads to an efficient algorithm to find the partition. More specifically, we show an deterministic algorithm that finds for any set $P \subseteq \mathbb{R}^d$ of $n$ points and any $k \in \{2, \dots, n\}$ in $O(nd \log k )$ time a partition of $P$ into $k$ subsets such that there is a ball of radius $O\left(\frac{k}{\sqrt{n}}\textrm{diam}(P)\right)$ intersecting the convex hull of each subset. A similar result holds also for the colorful version. To obtain our result, we generalize Sarkaria's tensor product constructions [Israel Journal Math., 1992] that reduces the Tverberg problem to the Colorful Caratheodory problem. By carefully choosing the vectors used in the tensor products, we implement the reduction in an efficient manner.Comment: A shorter version will appear at SoCG 202

No-Dimensional Tverberg Theorems and Algorithms

Tverberg’s theorem states that for any k≥2 and any set P⊂Rd of at least (d+1)(k−1)+1 points in d dimensions, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem of finding the partition lies in the complexity class CLS=PPAD∩PLS, but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa (SODA 2019) gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any n-point set P⊂Rd and any k∈{2,…,n} in O(nd⌈logk⌉) time a k-partition of P such that there is a ball of radius O((k/n−−√)diam(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, our result provides a remarkably efficient and simple new notion of approximation. Our main contribution is to generalize Sarkaria’s method (Israel Journal Math., 1992) to reduce the Tverberg problem to the colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem

Combinatorial properties of non-archimedean convex sets

We study combinatorial properties of convex sets over arbitrary valued fields. We demonstrate analogs of some classical results for convex sets over the reals (e.g. the fractional Helly theorem and B\'ar\'any's theorem on points in many simplices), along with some additional properties not satisfied by convex sets over the reals, including finite breadth and VC-dimension. These results are deduced from a simple combinatorial description of modules over the valuation ring in a spherically complete valued field.Comment: v.2: 27 pages; some minor corrections following referees' reports; added a brief discussion of the other notions of convexity in valued fields (Section 5.2) and connections to the study of abstract convexity spaces (Section 5.3); accepted to the Pacific Journal of Mathematic

Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend

LIPIcs

The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2