A theorem of Matoušek asserts that for any k ≥ 2, any set system whose shatter function is o(n^k) enjoys a fractional Helly theorem of order k: in the k-wise intersection hypergraph, positive density implies a linear-size clique. Kalai and Meshulam conjectured a generalization of that phenomenon to homological shatter functions. It was verified for set systems with bounded homological shatter functions and whose ground set has a forbidden homological minor (which includes ℝ^d by a homological analogue of the van Kampen-Flores theorem). We present two contributions to this line of research: - We study homological minors in certain manifolds (possibly with boundary), for which we prove analogues of the van Kampen-Flores theorem and of the Hanani-Tutte theorem. - We introduce graded analogues of the Radon and Helly numbers of set systems and relate their growth rate to the original parameters. This allows to extend the verification of the Kalai-Meshulam conjecture to sufficiently slowly growing homological shatter functions.

Abstract. 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.

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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.

Intersection patterns of convex sets in ℝ^d have the remarkable property that for d+1 ≤ k ≤ 𝓁, in any sufficiently large family of convex sets in ℝ^d, if a constant fraction of the k-element subfamilies have nonempty intersection, then a constant fraction of the 𝓁-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system ℱ in ℝ^d. Quantitatively, our bounds depend on how complicated the intersection of 𝓁 elements of ℱ can be, as measured by the maximum of the ⌈d/2⌉ 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, Matoušek and Nivasch to recast a simplicial complex as a homological minor of a cubical complex.

This dataset contains the reproducible research package for the preprint "Exact Value of M(8) and Sharp Bounds for Great-Circle Cell Expectations", addressing Oberwolfach Report 3/2024 "Open Problems in Discrete Geometry", Problem 8 (posed by Xavier Goaoc). Problem. Let S be a simple arrangement of n great circles on the sphere S^2 and choose a 2-dimensional cell c uniformly at random. Let S' be the circles that do not touch c, and let c' be the cell of the subarrangement S' that contains c. Define M(n) as the maximum, over all simple arrangements, of the expected number of edges of c'. Main results: - Theorem 1: exact value M(8) = 113/29, obtained by exhaustive enumeration over all 3,315 simple 8-point order types in the Aichholzer database; the extremal order-type index is 1026. - Theorem 2: for the regular near-pencil family, the closed-form expectation E[x] = (8 n^2 - 36 n) / (n^2 - n + 2) = 8 - O(1/n), giving M(n) >= 8 - O(1/n) for all n >= 6. - Theorem 3: general upper bound M(n) <= n(n-2)(n-3)/(n^2 - n + 2) = n - 4 + O(1/n^2) for all n >= 3. - Conjecture: the matching upper bound M(n) <= 8 + o(1) remains open; M(n) = 8 - o(1) is therefore a conjecture supported by numerical experiments. The archive includes Python scripts (MIT License), JSON data certificates and the Aichholzer order-type database (CC0 1.0 Universal), proof notes, review reports, and the preprint in PDF and Markdown form (CC-BY 4.0). Limitations: exact values are known only for n <= 8; the asymptotic upper bound is not proved.

We prove that for every d ≥ 2, deciding if a pure, d -dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d -dimensional, simplicial complex is k -decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible pure d -dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.

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 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.

We show that the union of n translates of a convex body in $$\mathbb {R}^3$$ can have $$\varTheta (n^3)$$ holes in the worst case, where a hole in a set X is a connected component of $$\mathbb {R}^3 \setminus X$$ . This refutes a 20-year-old conjecture. As a consequence, we also obtain improved lower bounds on the complexity of motion planning problems and of Voronoi diagrams with convex distance functions.

Given four congruent balls A,B,C,D in Rδ that have disjoint interior and admit a line that intersects them in the order ABCD, we show that the distance between the centers of consecutive balls is smaller than the distance between the centers of A and D. This allows us to give a new short proof that n interior-disjoint congruent balls admit at most three geometric permutations, two if n⩾7. We also make a conjecture that would imply that n⩾4 such balls admit at most two geometric permutations, and show that if the conjecture is false, then there is a counter-example that is algebraically highly degenerate.

We establish an upper bound on the smoothed complexity of convex hulls in R^d under uniform Euclidean (L^2) noise. Specifically, let {p_1^*, p_2^*, ..., p_n^*} be an arbitrary set of n points in the unit ball in R^d and let p_i = p_i^* + x_i, where x_1, x_2, ..., x_n are chosen independently from the unit ball of radius r. We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of {p_1, p_2, ..., p_n} is O(n^{2-4/(d+1)} (1+1/r)^{d-1}); the magnitude r of the noise may vary with n. For d=2 this bound improves to O(n^{2/3} (1+r^{-2/3})). We also analyze the expected complexity of the convex hull of L^2 and Gaussian perturbations of a nice sample of a sphere, giving a lower-bound for the smoothed complexity. We identify the different regimes in terms of the scale, as a function of n, and show that as the magnitude of the noise increases, that complexity varies monotonically for Gaussian noise but non-monotonically for L^2 noise.

The notion of limits of dense graphs was invented, among other reasons, to attack problems in extremal graph theory. It is straightforward to define limits of order types in analogy with limits of graphs, and this paper examines how to adapt to this setting two approaches developed to study limits of dense graphs. We first consider flag algebras, which were used to open various questions on graphs to mechanical solving via semidefinite programming. We define flag algebras of order types, and use them to obtain, via the semidefinite method, new lower bounds on the density of 5- or 6-tuples in convex position in arbitrary point sets, as well as some inequalities expressing the difficulty of sampling order types uniformly. We next consider graphons, a representation of limits of dense graphs that enable their study by continuous probabilistic or analytic methods. We investigate how planar measures fare as a candidate analogue of graphons for limits of order types. We show that the map sending a measure to its associated limit is continuous and, if restricted to uniform measures on compact convex sets, a homeomorphism. We prove, however, that this map is not surjective. Finally, we examine a limit of order types similar to classical constructions in combinatorial geometry (Erdos-Szekeres, Horton...) and show that it cannot be represented by any somewhere regular measure; we analyze this example via an analogue of Sylvester's problem on the probability that k random points are in convex position.

The fact that the complete graph K_5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M), where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2. Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van Kampen--Flores theorem. In the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.

Average-case analysis of data-structures or algorithms is commonly used in computational geometry when the, more classical, worst-case analysis is deemed overly pessimistic. Since these analyses are often intricate, the models of random geometric data that can be handled are often simplistic and far from "realistic inputs". We present a new simple scheme for the analysis of geometric structures. While this scheme only produces results up to a polylog factor, it is much simpler to apply than the classical techniques and therefore succeeds in analyzing new input distributions related to smoothed complexity analysis.

We consider the problem of computing shortest paths having curvature at most one almost everywhere and visiting a sequence of $n$ points in the plane in a given order. This problem is a subproblem of the Dubins traveling salesman problem and also arises naturally in path planning for point car-like robots in the presence of polygonal obstacles. We show that when consecutive waypoints are a distance of at least four apart, this question reduces to a family of convex optimization problems over polyhedra in $\mathbb{R}^n$.

The nerve of a family of sets is a simplicial complex that records the intersection pattern of its subfamilies. Nerves are widely used in computational geometry and topology, because the nerve theorem guarantees that the nerve of a family of geometric objects has the same topology as the union of the objects, if they form a good cover. In this paper, we relax the good cover assumption to the case where each subfamily intersects in a disjoint union of possibly several homology cells, and we prove a generalization of the nerve theorem in this framework, using spectral sequences from algebraic topology. We then deduce a new topological Helly-type theorem that unifies previous results of Amenta, Kalai and Meshulam, and Matousek. This Helly-type theorem is used to (re)prove, in a unified way, bounds on transversal Helly numbers in geometric transversal theory.

This paper presents a complete analytical characterization of a large class of central and non-central imaging devices dubbed linear cameras by Ponce. Pajdla has shown that a subset of these, the oblique cameras, can be modelled by a certain type of linear map. We give here a full tabulation of all admissible maps that induce cameras in the general sense of Grossberg and Nayar, and show that these cameras are exactly the linear ones. Combining these two models with a new notion of intrinsic parameters and normalized coordinates for linear cameras allows us to give simple analytical formulas for direct and inverse projections. We also show that the epipolar geometry of any two linear cameras can be characterized by a fundamental matrix whose size is at most 6 × 6 when the cameras are uncalibrated, or by an essential matrix of size at most 4 × 4 when their internal parameters are known. Similar results hold for trinocular constraints.

A straight-line drawing δ of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G,δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-drawing problem. Further we define fix(G,δ)=n−shift(G,δ) to be the maximum number of vertices of a planar n-vertex graph G that can be fixed when untangling δ. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log\log n}$ vertices when untangling a drawing of an n-vertex graph G. If G is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$ vertices. On the other hand, we construct, for arbitrarily large n, an n-vertex planar graph G and a drawing δ G of G with $\ensuremath {\mathrm {fix}}(G,\delta_{G})\leq \sqrt{n-2}+1$ and an n-vertex outerplanar graph H and a drawing δ H of H with $\ensuremath {\mathrm {fix}}(H,\delta_{H})\leq2\sqrt{n-1}+1$ . Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

Let F ∪ {U} be a collection of convex sets in Rd such that F covers U. We show that if the elements of F and U have comparable size, in the sense that each contains a ball of radius r and is contained in a ball of radius R for some fixed r and R, then for any ε > 0 there exists Hε ⊂ F, whose size |Hε| is polynomial in 1/ε and independent of |F|, that covers U except for a volume of at most ε. The size of the smallest such subset depends on the geometry of the elements of F; specifically, we prove that it is O(1/ε) when F consists of axis-parallel unit squares in the plane and Õ(ε1--d/2) when F consists of unit balls in Rd (here, Õ(n) means O(n log n) for some constant), and that these bounds are, in the worst-case, tight up to the logarithmic factors.

We prove that the set of directions of lines intersecting three disjoint balls in R3 in a given order is a strictly convex subset of S2. We then generalize this result to n disjoint balls in Rd. As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.

Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of three-dimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in $\mathbb{R}^3$ with a total of n edges consists of $\Theta(n^2)$ connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrarily degenerate scenes. More generally, we show that a set of k possibly intersecting convex polyhedra with a total of n edges admits, in the worst case, $\Theta(n^2k^2)$ connected components of maximal free line segments tangent to at least four polytopes. Furthermore, these bounds also hold for possibly occluded lines rather than maximal free line segments. Finally, we present an $O(n^2 k^2 \log n)$ time and $O(nk^2)$ space algorithm that, given a scene of k possibly intersecting convex polyhedra, computes all the minimal free line segments that are tangent to any four of the polytopes and are isolated transversals to the set of edges they intersect; in particular, we compute at least one line segment per connected component of tangent lines.

Let S be an ordered set of disjoint unit spheres in R3 We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal forS.

We prove that the lines tangent to four possibly intersecting convex polyhedra in ℝ3 with n edges in total form Θ(n2) connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrary degenerate scenes. More generally, we show that a set of kconvex polyhedra with a total of n edges admits, in the worst case, Θ(n2k2)connected components of (possibly occluded) lines tangent to any four of these polyhedra. We also show a lower bound of Ω(n2k2) on the number of non-occluded maximal line segments tangent to any four of these k convex polyhedra.

In this paper, we show that, amongst n uniformly distributed unit balls in $\mathbb{R}^3$, the expected number of maximal nonoccluded line segments tangent to four balls is linear. Using our techniques we show a linear bound on the expected size of the visibility complex, a data structure encoding the visibility information of a scene, providing evidence that the storage requirement for this data structure is not necessarily prohibitive. These results significantly improve the best previously known bounds of $O(n^{8/3})$ [F. Durand, G. Drettakis, and C. Puech, {ACM Transactions on Graphics}, 21 (2002), pp. 176-206]. Our results generalize in various directions. We show that the linear bound on the expected number of maximal nonoccluded line segments that are not too close to the boundary of the scene and tangent to four unit balls extends to balls of various but bounded radii, to polyhedra of bounded aspect ratio, and even to nonfat three-dimensional objects such as polygons of bounded aspect ratio. We also prove that our results extend to other distributions such as the Poisson distribution. Finally, we indicate how our probabilistic analysis provides new insight on the expected size of other global visibility data structures, notably the aspect graph.