In this paper we study generalizations of classical results on intersection patterns of set systems in ℝ^d, such as the fractional Helly theorem or the (p,q)-theorem, in the setting of arbitrary triangulable spaces with a forbidden homological minor. Given a simplicial complex K and an integer b, we say that a family ℱ of subcomplexes of some simplicial complex X is a (K,b)-free cover if (i) K is a forbidden homological minor of X, and (ii) the jth reduced Betti number _j(⋂_S∈𝒢S,ℤ_2) is strictly less than b for all 0≤ j < K and all nonempty subfamilies 𝒢⊆ℱ. We show that for every K and b, the fractional Helly number of a (K,b)-free cover is at most μ(K)+1, where μ(K) is the maximum sum of the dimensions of two disjoint faces in K. This implies that the assertion of the (p,q)-theorem holds for every p ≥ q > μ(K) and every (K,b)-free cover ℱ. For b=1 and a suitable K this recovers the original (p,q)-theorem and its generalization to good covers. Interestingly, our results show that that the range of parameters (p,q) for which the (p,q)-theorem holds is independent of b. Our proofs use Ramsey-type arguments combined with the notion of stair convexity of Bukh et al. to construct (forbidden) homological minors in certain cubical complexes.

Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.

We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets (which we rephrase as modules), and adapts in some sense the modular decomposition of graphs in the world of chirotopes. The associated tree always exists and is unique up to some appropriate constraints. We also show how to compute the number of triangulations of a chirotope efficiently, starting from its tree and the (weighted) numbers of triangulations of its parts.

Computational Geometry contains a selection of papers presented at the 38th International Symposium on Computational Geometry (SoCG 2022) held in Berlin, on June 7-10, 2022, as part of the Computational Geometry Week (CGWeek 2022).Altogether, 174 papers were submitted to SoCG2022, among which the program committee selected 64 papers to be presented at the conference.The ten papers in this issue were invited because they received a strong support within the program committee and provide a sample of the excellent research presented at the conference in a broad range of topics.The papers from this issue were invited, submitted, reviewed, and revised according to the usual high standards of D&CG.We thank the authors of all the submitted papers for revising and polishing their work.We are very grateful to the anonymous referees for their dedication, expertise,

We investigate a number of questions, problems, and conjectures related to geometric transversal theory. Among our results we disprove a conjecture of Bárány and Kalai regarding weak ε -nets for k -flats and convex sets in ℝ^d , and we prove a conjecture of Arocha, Bracho, and Montejano regarding a colorful version of the Goodman–Pollack–Wenger transversal theorem. We also investigate the connected components of the space of line transversals to pairwise disjoint convex sets in ℝ^3 , and we extend a theorem of Karasev and Montejano regarding colorful intersections and k -transversals.

The authors first study the degree of standard procedures for determining the number of line transversals to four lines or four segments in 3D [cf. H. Brönnimann, H. Everett, S. Lazard, F. Sottile and S. Whitesides , Discrete Comput. Geom. 34, No. 3, 381–390 (2005; Zbl 1083.52003)]. They also consider the predicate for determining whether a minimal segment transversal to four line segments is intersected by a triangle. These predicators are ubiquitous in 3D visibility problems The predicate for ordinary planes through two fixed points, each plane containing a third rational point or a line transversal to four segments or lines is also studied [cf. H. Brönnimann, O. Devillers, V. Dujmović, H. Everett, M. Glisse, X. Goaoc, S. Lazard, H.-S. Na , and S. Whitesides SIAM J. Comput. 37, No. 2, 522–551 (2007; Zbl 1138.65019)].

We prove that there exist no weak $\varepsilon$-nets of constant size for lines and convex sets in $\mathbb{R}^d$.

We prove that for any set F of n≥ 2 pairwise disjoint open convex sets in ℝ^3, the connected components of the set of lines intersecting every member of F are contractible. The same result holds for directed lines.

A first public introduction to these questions gave way to a lively discussion both in the open forum and on the associated discord server where several concerns were raised around these issues. Two votes were organized at the business meeting, with about 120 votes cast in each. The first vote enquired about the creation of a task force to investigate some form of double blind; the results were strongly in favor, so a task force was created. The second vote showed support for PC submission so the Steering committee included this question in the mandate of the task force. In the discussion, there was a general agreement that the process should be as fair as possible, and that double-blind reviews had the potential to increase fairness. The main concerns raised deal with two trade-offs to be made, and which underlie any double-blind system: prevention of author uncovering VS knowledge dissemination, and bias reduction VS vulnerability to fraud. Let us stress, however, that any review process is at heart a trust-based system, not designed to be fully resistant again fraud or abuses. This includes the current single-blind system as well as any double-blind system that we consider. The question is really one of setting a trade-off between various adverse effects, which should not be considered in binary terms. It is worth noting that this discussion is happening more broadly in the field of computer science; see http://double-blind.org/ for a listing of major computer science conferences and their current status with regards to double blind reviewing.

Intersection patterns of convex sets in $\mathbb{R}^d$ have the remarkable property that for $d+1 \le k \le \ell$, in any sufficiently large family of convex sets in $\mathbb{R}^d$, if a constant fraction of the $k$-element subfamilies have nonempty intersection, then a constant fraction of the $\ell$-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system $\mathcal{F}$ in $\mathbb{R}^d$. Quantitatively, our bounds depend on how complicated the intersection of $\ell$ elements of $\mathcal{F}$ can be, as measured by the sum of the $\lceil\frac{d}2\rceil$ first Betti numbers. As an application, we improve the fractional Helly number of set systems with bounded topological complexity due to the third author, from a Ramsey number down to $d+1$. We also shed some light on a conjecture of Kalai and Meshulam on intersection patterns of sets with bounded homological VC dimension. A key ingredient in our proof is the use of the stair convexity of Bukh, Matou\v{s}ek and Nivash to recast a simplicial complex as a homological minor of a cubical complex.

We examine how the measure and the number of vertices of the convex hull of a random sample of n points from an arbitrary probability measure in 𝐑^d relates to the wet part of that measure. This extends classical results for the uniform distribution from a convex set [Bárány and Larman 1988]. The lower bound of Bárány and Larman continues to hold in the general setting, but the upper bound must be relaxed by a factor of log n. We show by an example that this is tight.

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n -point order type, chosen uniformly at random from all such order types, is on average 4+ o (1). For labeled order types, this number has average \(4- \mbox{$\frac{8}{n^2 - n +2}$}\) and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2 d + o (1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős-Szekeres theorem: for any fixed k , as n → ∞, a proportion 1 - O (1/ n ) of the n -point simple order types contain a triangle enclosing a convex k -chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A 4 , S 4 , or A 5 (and each case is possible). These are the finite subgroups of SO (3) and our proof follows the lines of their characterization by Felix Klein.

We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in $\mathbb{R}^3$. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be "lifted" to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations.

We study how a single value of the shatter function of a set system restricts its asymptotic growth. Along the way, we refute a conjecture of Bondy and Hajnal which generalizes Sauer's Lemma.

We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath\'eodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as game theory, graph theory, mathematical optimization, computational geometry, etc.

We consider incidences among colored sets of lines in $\mathbb{R}^d$ and examine whether the existence of certain concurrences between lines of $k$ colors force the existence of at least one concurrence between lines of $k+1$ colors. This question is relevant for problems in 3D reconstruction in computer vision.