Helly's Theorem Research Articles

Two remarks on properties of functions of bounded variation

In terms of variations, a sufficient condition for the uniform convergence of sequences of continuous functions is proved. Using this result, we obtain an addition to the classical Helly theorem on the selection of convergent sequences of functions with uniformly bounded variations in the case when the limit function is continuous. Also, by using an example we show that the condition of continuous differentiability of a function, ensuring the differentiability of its variation with the variable upper limit, is in a certain sense sharp.

A selection theorem for essentially regulated functions of several variables

Helly's selection theorem provides a criterion for compactness of sets of single-variable functions with bounded pointwise variation. Fraňková has given a proper extension of Helly's theorem to the setting of single-variable regulated functions. We show how a similar approach yields a natural notion of regulated functions of several variables, together with an extension of the standard compactness criterion for multi-variable functions of bounded variation.

Extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes

Abstract We present extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes, providing a common generalization to the matroidal versions of the theorem due to Kalai and Meshulam, the ‘very colorful’ Helly theorem introduced by Arocha, Bárány, Bracho, Fabila and Montejano and the ‘semi-intersecting’ colorful Helly theorem proved by Montejano and Karasev. As an application, we obtain the following extension of Tverberg’s theorem: Let A be a finite set of points in ${\mathbb R}^d$ with $|A|>(r-1)(d+1)$ . Then, there exist a partition $A_1,\ldots ,A_r$ of A and a subset $B\subset A$ of size $(r-1)(d+1)$ such that $\cap _{i=1}^r \operatorname {\mathrm {\text {conv}}}( (B\cup \{p\})\cap A_i)\neq \emptyset $ for all $p\in A\setminus B$ . That is, we obtain a partition of A into r parts that remains a Tverberg partition even after removing all but one arbitrary point from $A\setminus B$ .

Weak Saturation of Multipartite Hypergraphs

Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F, G is called weaklyH-saturated in F if the edges in E(F)setminus E(G) admit an ordering e_1,ldots , e_k so that for all iin [k] the hypergraph Gcup {e_1,ldots ,e_i} contains an isomorphic copy of H which in turn contains the edge e_i. The weak saturation number of H in F is the smallest size of an H-weakly saturated subgraph of F. Weak saturation was introduced by Bollobás in 1968, but despite decades of study our understanding of it is still limited. The main difficulty lies in proving lower bounds on weak saturation numbers, which typically withstands combinatorial methods and requires arguments of algebraic or geometrical nature. In our main contribution in this paper we determine exactly the weak saturation number of complete multipartite q-graphs in the directed setting, for any choice of parameters. This generalizes a theorem of Alon from 1985. Our proof combines the exterior algebra approach from the works of Kalai with the use of the colorful exterior algebra motivated by the recent work of Bulavka, Goodarzi and Tancer on the colorful fractional Helly theorem. In our second contribution answering a question of Kronenberg, Martins and Morrison, we establish a link between weak saturation numbers of bipartite graphs in the clique versus in a complete bipartite host graph. In a similar fashion we asymptotically determine the weak saturation number of any complete q-partite q-graph in the clique, generalizing another result of Kronenberg et al.

Leray Numbers of Tolerance Complexes

Let K be a simplicial complex on vertex set V. K is called d-Leray if the homology groups of any induced subcomplex of K are trivial in dimensions d and higher. K is called d-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most d that is contained in a unique maximal face. Motivated by results of Eckhoff and of Montejano and Oliveros on “tolerant” versions of Helly’s theorem, we define the t-tolerance complex of K, {mathcal {T}}_{t}(K), as the simplicial complex on vertex set V whose simplices are formed as the union of a simplex in K and a set of size at most t. We prove that for any d and t there exists a positive integer h(t, d) such that, for every d-collapsible complex K, the t-tolerance complex {mathcal {T}}_t(K) is h(t, d)-Leray. As an application, we present some new tolerant versions of the colorful Helly theorem.

Combinatorial properties of nonarchimedean 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.

Fractional Helly Theorem for Cartesian Products of Convex Sets

Helly’s theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Bárány and Kalai, and independently Lew, we generalize Eckhoff’s result to Cartesian products of convex sets in all dimensions. In particular, we prove that given $$\alpha \in (1-1/t^d,1]$$ and a finite family $${\mathcal {F}}$$ of Cartesian products of convex sets $$\prod _{i\in [t]}A_i$$ in $${\mathbb {R}}^{td}$$ with $$A_i\subset {\mathbb {R}}^d$$ , if at least $$\alpha $$ -fraction of the $$(d+1)$$ -tuples in $${\mathcal {F}}$$ are intersecting, then at least $$(1-(t^d(1-\alpha ))^{1/(d+1)})$$ -fraction of sets in $${\mathcal {F}}$$ are intersecting. This is a special case of a more general result on intersections of d-Leray complexes. We also provide a construction showing that our result on d-Leray complexes is optimal. Interestingly, the extremal example is representable as a family of Cartesian products of convex sets, implying that the bound $$\alpha >1-1/t^d$$ and the fraction $$(1-(t^d(1-\alpha ))^{1/(d+1)})$$ above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in $${\mathbb {R}}^d$$ does not have $$(p,d+1)$$ -condition for sublinear p, that is, it contains a linear-size subfamily with no intersecting $$(d+1)$$ -tuple. Inspired by this, we give constructions showing that, somewhat surprisingly, imposing an additional $$(p,d+1)$$ -condition has a negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable.

Stable Bounds on the Duality Gap of Separable Nonconvex Optimization Problems

The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland show that this produces an a priori bound on the duality gap of separable nonconvex optimization problems involving finite sums. This bound is highly conservative and depends on unstable quantities; we relax it in several directions to show that nonconvexity can have a much milder impact on finite sum minimization problems, such as empirical risk minimization and multitask classification. As a byproduct, we show a new version of Maurey’s classical approximate Carathéodory lemma where we sample a significant fraction of the coefficients, without replacement, as well as a result on sampling constraints using an approximate Helly theorem, both of independent interest. Funding: A. d’Aspremont acknowledges support from the French government under management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program [Grant ANR-19-P3IA-0001] (PRAIRIE 3IA Institute), the Machine Learning and Optimisation Joint Research Initiative with the Fonds AXA pour la Recherche and Kamet Ventures, as well as a Google focused award.

Quantitative Fractional Helly and [formula omitted]-Theorems

We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main result is a quantitative version of the Fractional Helly Theorem of Katchalski and Liu, the second one is a quantitative version of the (p,q)-Theorem of Alon and Kleitman.

Eckhoff's Problem on Convex Sets in the Plane

Eckhoff proposed a combinatorial version of the classical Hadwiger–Debrunner $(p,q)$-problems as follows. Let ${\cal F}$ be a finite family of convex sets in the plane and let $m\geqslant 1$ be an integer. If among every ${m+2\choose 2}$ members of ${\cal F}$ all but at most $m-1$ members have a common point, then there is a common point for all but at most $m-1$ members of ${\cal F}$. The claim is an extension of Helly's theorem ($m=1$). The case $m=2$ was verified by Nadler and by Perles. Here we show that Eckhoff 's conjecture follows from an old conjecture due to Szemerédi and Petruska concerning $3$-uniform hypergraphs. This conjecture is still open in general; its solution for a few special cases answers Eckhoff's problem for $m=3,4$. A new proof for the case $m=2$ is also presented.

Radon Numbers Grow Linearly

Define the k-th Radon number r_k of a convexity space as the smallest number (if it exists) for which any set of r_k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that r_k grows linearly, i.e., r_kle c(r_2)cdot k.

RELATIVE LERAY NUMBERS VIA SPECTRAL SEQUENCES

Let F be a fixed field and let X be a simplicial complex on the vertex set V. The Leray number L ( X ; F ) is the minimal d such that for all i ⩾ d and S ⊂ V , the induced complex X [ S ] satisfies H ∼ i ( X [ S ] ; F ) = 0 . Leray numbers play a role in formulating and proving topological Helly-type theorems. For two complexes X , Y on the same vertex set V, define the relative Leray number L Y ( X ; F ) as the minimal d such that H ∼ i ( X [ V ∖ τ ] ; F ) = 0 for all i ⩾ d and τ ∈ Y . In this paper we extend the topological colorful Helly theorem to the relative setting. Our main tool is a spectral sequence for the intersection of complexes indexed by a geometric lattice.

Discrete quantitative Helly-type theorems with boxes

Research on Helly-type theorems in combinatorial convex geometry has produced volumetric versions of Helly's theorem using witness sets and quantitative extensions of Doignon's theorem. This paper combines these philosophies and presents quantitative Helly-type theorems for the integer lattice with axis-parallel boxes as witness sets. Our main result shows that, while quantitative Helly numbers for the integer lattice grow polynomially in each fixed dimension, their variants with boxes as witness sets are uniformly bounded. We prove several colorful and fractional variations on this theorem. We also prove that the Helly number for A×A⊆R2 need not be finite even when A⊆Z is a syndetic set.

Radon numbers and the fractional Helly theorem

A basic measure of the combinatorial complexity of a convexity space is its Radon number. In this paper we answer a question of Kalai, by showing a fractional Helly theorem for convexity spaces with bounded Radon number. As a consequence we also get a weak ε-net theorem for convexity spaces with bounded Radon number. This answers a question of Bukh and extends a recent result of Moran and Yehudayoff.

Median pretrees and functions of bounded variation

We introduce functions of bounded variation on median algebras and study some properties for median pretrees. We show that if X is a compact median pretree (e.g., a dendron) in its shadow topology then every function f:X→R of bounded variation has the point of continuity property (Baire 1, if X, in addition, is metrizable). We prove a generalized version of Helly's selection theorem for a sequence of functions with total bounded variation defined on a Polish median pretree X.

Large Cliques in Hypergraphs with Forbidden Substructures

A result due to Gyarfas, Hubenko, and Solymosi (answering a question of Erdős) states that if a graph G on n vertices does not contain K2,2 as an induced subgraph yet has at least $$c\left(\begin{array}{c}n 2\end{array}\right)$$ edges, then G has a complete subgraph on at least $$\frac{c^2}{10}n$$ vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced K2,2 which allows us to generalize their result to k-uniform hypergraphs. Our result also has an interesting consequence in discrete geometry. In particular, it implies that the fractional Helly theorem can be derived as a purely combinatorial consequence of the colorful Helly theorem.

An Analog of the Bondareva-Shapley Theorem I. The Non-Emptiness of the Core of a Fuzzy Game

This paper deals with a generalization of the famous Bondareva-Shapley theorem [1, 9] on the core of TU cooperative games to the case of fuzzy blocking. The suggested approach is based on the concept of a balanced collection of fuzzy coalitions. Introduced by the author, this extension of the classical balanced collection of standard coalitions yields a natural analog of balancedness for the so-called fuzzy TU cooperative games. As established below, the general balancedness is a necessary and sufficient condition for the non-emptiness of the core of fuzzy TU cooperative games. The non-emptiness criterion of the core is further refined using the classical Helly's theorem on the intersection of convex sets. The S*-representation of a fuzzy game is studied, which simplifies the existence conditions for non-blocking imputations of this game in a series of cases.

Further Consequences of the Colorful Helly Hypothesis

Let \(\mathcal {F}\) be a family of convex sets in \({\mathbb {R}}^d,\) which are colored with \(d+1\) colors. We say that \(\mathcal {F}\) satisfies the Colorful Helly Property if every rainbow selection of \(d+1\) sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family \(\mathcal {F}\) there is a color class \(\mathcal {F}_i\subset \mathcal {F},\) for \(1\le i\le d+1,\) whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension \(d\ge 2\) there exist numbers f(d) and g(d) with the following property: either one can find an additional color class whose sets can be pierced by f(d) points, or all the sets in \(\mathcal {F}\) can be crossed by g(d) lines.

3D unsteady Navier–Stokes problem with memory and subdifferential boundary condition

We consider a mathematical model which describes the motion of a 3D unsteady fluid flow governed by the Navier–Stokes system, and subjected to mixed boundary conditions with a given velocity on one part of the boundary and nonlinear slip conditions with a memory term reminiscent of Coulomb's friction law on the other part. We establish first some regularity properties and estimates for a simplified model. Then we prove the existence of a solution to our problem by using a successive approximation technique and compactness arguments based on Helly's theorem for the velocity field.

A Helly theorem for an enlarged class of orthogonally starshaped sets in $$\mathbb {R}^d$$ R d

Consider a finite family $$\mathcal {C}$$ of distinct boxes in $$\mathbb {R}^d$$ , with G the intersection graph of $$\mathcal {C}$$ and with $$S \equiv \cup \{C : C \, \hbox {in} \, \mathcal {C}\}$$ . Assume that G is connected and, for each block B of G, assume that the corresponding members of $$\mathcal {C}$$ have a staircase convex union D(B). When this occurs, we say that the orthogonal polytope S has suitable properties. Now let $$\mathbf{S}$$ be a finite family of orthogonal polytopes such that, for every nonempty subfamily $$\mathbf{S}^{'}$$ of $$\mathbf{S}$$ , the corresponding intersection $$S^{'} \equiv \cap \{S : S \, \hbox {in} \, \mathbf{S}^{'} \}$$ (if nonempty) has suitable properties and preserves transitions between certain D(B) sets. If every $$d +1$$ (not necessarily distinct) members of $$\mathbf{S}$$ meet in a (nonempty) staircase starshaped set, then $$S_0 \equiv \cap \{S : S \, \hbox {in} \, \mathbf{S} \}$$ is nonempty and staircase starshaped as well.