Many Collinear k-Tuples with no k+1 Collinear Points

For every $k>3$ k > 3 , we give a construction of planar point sets with many collinear $k$ k -tuples and no collinear $(k+1)$ ( k + 1 ) -tuples. We show that there are $n_0=n_0(k)$ n 0 = n 0 ( k ) and $c=c(k)$ c = c ( k ) such that if $n\ge n_0$ n ≥ n 0 , then there exists a set of $n$ n points in the plane that does not contain $k+1$ k + 1 points on a line, but it contains at least $n^{2-({c}/{\sqrt{\log n}})}$ n 2 - ( c / log n ) collinear $k$ k -tuples of points. Thus, we significantly improve the previously best known lower bound for the largest number of collinear $k$ k -tuples in such a set, and get reasonably close to the trivial upper bound $O(n^2)$ O ( n 2 ) .     

Polytopal Affine Semigroups with Holes Deep Inside

Given a positive integer $k$ k , we construct a lattice $3$ 3 -simplex $P$ P with the following property: The affine semigroup $Q_P$ Q P associated to $P$ P is not normal, and every element $q \in \overline{Q}_P \setminus Q_P$ q ∈ Q ¯ P ? Q P has lattice distance at least $k$ k above every facet of $Q_P$ Q P .     

A Question from a Famous Paper of Erdős

Given a convex body $K$ K , consider the smallest number $N$ N so that there is a point $P\in \partial K$ P ∈ ? K such that every circle centred at $P$ P intersects $\partial K$ ? K in at most $N$ N points. In 1946 Erd?s conjectured that $N=2$ N = 2 for all $K$ K , but there are convex bodies for which this is not the case. As far as we know there is no known global upper bound. We show that no convex body has $N=\infty $ N = ∞ and that there are convex bodies for which $N = 6$ N = 6 .     

Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

For a polyhedron $P$ P let $B(P)$ B ( P ) denote the polytopal complex that is formed by all bounded faces of $P$ P . If $P$ P is the intersection of $n$ n halfspaces in $\mathbb R ^D$ R D , but the maximum dimension $d$ d of any face in $B(P)$ B ( P ) is much smaller, we show that the combinatorial complexity of $P$ P cannot be too high; in particular, that it is independent of $D$ D . We show that the number of vertices of $P$ P is $O(n^d)$ O ( n d ) and the total number of bounded faces of the polyhedron is $O(n^{d^2})$ O ( n d 2 ) . For inputs in general position the number of bounded faces is $O(n^d)$ O ( n d ) . We show that for certain specific values of $d$ d and $D$ D , our bounds are tight. For any fixed $d$ d , we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a number of linear programs that is polynomial in  $n$ n .     

Maximal Equilateral Sets

A subset of a normed space $X$ X is called equilateral if the distance between any two points is the same. Let $m(X)$ m ( X ) be the smallest possible size of an equilateral subset of $X$ X maximal with respect to inclusion. We first observe that Petty’s construction of a $d$ d - $X$ X of any finite dimension $d\ge 4$ d ≥ 4 with $m(X)=4$ m ( X ) = 4 can be generalised to give $m(X\oplus _1\mathbb R )=4$ m ( X ⊕ 1 R ) = 4 for any $X$ X of dimension at least $2$ 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that for any set $\Gamma $ Γ , $m(\ell _p(\Gamma ))$ m ( ? p ( Γ ) ) is finite and bounded above by a function of $p$ p , for all $1\le p<2$ 1 ≤ p < 2 . Also, for all $p\in [1,\infty )$ p ∈ [ 1 , ∞ ) and $d\in \mathbb N $ d ∈ N there exists $c=c(p,d)>1$ c = c ( p , d ) > 1 such that $m(X)\le d+1$ m ( X ) ≤ d + 1 for all $d$ d - $X$ X with Banach–Mazur distance less than $c$ c from $\ell _p^d$ ? p d . Using Brouwer’s fixed-point theorem we show that $m(X)\le d+1$ m ( X ) ≤ d + 1 for all $d$ d - $X$ X with Banach–Mazur distance less than $3/2$ 3 / 2 from $\ell _\infty ^d$ ? ∞ d . A graph-theoretical argument furthermore shows that $m(\ell _\infty ^d)=d+1$ m ( ? ∞ d ) = d + 1 . The above results lead us to conjecture that $m(X)\le 1+\dim X$ m ( X ) ≤ 1 + dim X for all finite-normed spaces $X$ X .     

The 2-Page Crossing Number of K_{n}

Around 1958, Hill described how to draw the complete graph $K_n$ K n with $$\begin{aligned} Z(n) :=\frac{1}{4}\Big \lfloor \frac{n}{2}\Big \rfloor \Big \lfloor \frac{n-1}{2}\Big \rfloor \Big \lfloor \frac{n-2}{2}\Big \rfloor \Big \lfloor \frac{n-3}{2}\Big \rfloor \end{aligned}$$ Z ( n ) : = 1 4 ? n 2 ? ? n ? 1 2 ? ? n ? 2 2 ? ? n ? 3 2 ? crossings, and conjectured that the crossing number ${{\mathrm{cr}}}(K_{n})$ cr ( K n ) of $K_n$ K n is exactly $Z(n)$ Z ( n ) . This is also known as Guy’s conjecture as he later popularized it. Towards the end of the century, substantially different drawings of $K_{n}$ K n with $Z(n)$ Z ( n ) crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line $\ell $ ? (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by  $\ell $ ? . The 2-page crossing number of $K_{n} $ K n , denoted by $\nu _{2}(K_{n})$ ν 2 ( K n ) , is the minimum number of crossings determined by a 2-page book drawing of $K_{n}$ K n . Since ${{\mathrm{cr}}}(K_{n}) \le \nu _{2}(K_{n})$ cr ( K n ) ≤ ν 2 ( K n ) and $\nu _{2}(K_{n}) \le Z(n)$ ν 2 ( K n ) ≤ Z ( n ) , a natural step towards Hill’s Conjecture is the weaker conjecture $\nu _{2}(K_{n}) = Z(n)$ ν 2 ( K n ) = Z ( n ) , popularized by Vrt’o. In this paper we develop a new technique to investigate crossings in drawings of $K_{n}$ K n , and use it to prove that $\nu _{2}(K_{n}) = Z(n) $ ν 2 ( K n ) = Z ( n ) . To this end, we extend the inherent geometric definition of $k$ k -edges for finite sets of points in the plane to topological drawings of $K_{n}$ K n . We also introduce the concept of ${\le }{\le }k$ ≤ ≤ k -edges as a useful generalization of ${\le }k$ ≤ k -edges and extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of $K_{n}$ K n in terms of its number of ${\le }k$ ≤ k -edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of $K_{n}$ K n and show that, up to equivalence, they are unique for $n$ n even, but that there exist an exponential number of non homeomorphic such drawings for $n$ n odd.     

On Sets Defining Few Ordinary Lines

Let $P$ P be a set of $n$ n points in the plane, not all on a line. We show that if $n$ n is large then there are at least $n/2$ n / 2 ordinary lines, that is to say lines passing through exactly two points of $P$ P . This confirms, for large $n$ n , a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $n-C$ n - C ordinary lines for some absolute constant $C$ C . We also solve, for large $n$ n , the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of $P$ P . Underlying these results is a structure theorem which states that if $P$ P has at most $Kn$ K n ordinary lines then all but O(K) points of $P$ P lie on a cubic curve, if $n$ n is sufficiently large depending on $K$ K .     

Tilings of Parallelograms with Similar Right Triangles

We say that a triangle $T$ T tiles the polygon $\mathcal A $ A if $\mathcal A $ A can be decomposed into finitely many non-overlapping triangles similar to $T$ T . A tiling is called regular if there are two angles of the triangles, say $\alpha $ α and $\beta $ β , such that at each vertex $V$ V of the tiling the number of triangles having $V$ V as a vertex and having angle $\alpha $ α at $V$ V is the same as the number of triangles having angle $\beta $ β at $V$ V . Otherwise the tiling is called irregular. Let $\mathcal P (\delta )$ P ( δ ) be a parallelogram with acute angle $\delta $ δ . In this paper we prove that if the parallelogram $\mathcal P (\delta )$ P ( δ ) is tiled with similar triangles of angles $(\alpha , \beta , \pi /2)$ ( α , β , π / 2 ) , then $(\alpha , \beta )=(\delta , \pi /2-\delta )$ ( α , β ) = ( δ , π / 2 - δ ) or $(\alpha , \beta )=(\delta /2, \pi /2-\delta /2)$ ( α , β ) = ( δ / 2 , π / 2 - δ / 2 ) , and if the tiling is regular, then only the first case can occur.     

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ X in $\mathbb{R }^2$ R 2 that is not an axis-aligned rectangle and for any positive integer $k$ k produces a family $\mathcal{F }$ F of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$ X , such that no three sets in $\mathcal{F }$ F pairwise intersect and $\chi (\mathcal{F })>k$ χ ( F ) > k . This provides a negative answer to a question of Gyárfás and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.     

Affine Mirković-Vilonen polytopes

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra \(\mathfrak{g}\) has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given \(\mathfrak{g}\) , there is a limit crystal, usually denoted by B(?∞), which contains all the other crystals. When \(\mathfrak{g}\) is finite dimensional, a convex polytope, called the Mirkovi?-Vilonen polytope, can be associated to each element in B(?∞). This polytope sits in the dual space of a Cartan subalgebra of \(\mathfrak{g}\) , and its edges are parallel to the roots of \(\mathfrak{g}\) . In this paper, we generalize this construction to the case where \(\mathfrak{g}\) is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(?∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as  \(\mathfrak{g}\) . The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category \(\Lambda \text {\upshape -}\mathrm {mod}\) . The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.     

Recovering an Homogeneous Polynomial from Moments of Its Level Set

Let $\mathbf{K }:=\left\{ \mathbf{x }: g(\mathbf{x })\le 1\right\} $ K : = x : g ( x ) ≤ 1 be the compact (and not necessarily convex) sub-level set of some homogeneous polynomial $g$ g . Assume that the only knowledge about $\mathbf{K }$ K is the degree of $g$ g as well as the moments of the Lebesgue measure on $\mathbf{K }$ K up to order $2d$ 2 d . Then the vector of coefficients of $g$ g is the solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order $2d$ 2 d of the Lebesgue measure on $\mathbf{K }$ K encode all information on the homogeneous polynomial $g$ g that defines $\mathbf{K }$ K (in fact, only moments of order $d$ d and $2d$ 2 d are needed).     

Boundaries of Disk-Like Self-affine Tiles

Let $T:= T(A, \mathcal{D })$ T : = T ( A , D ) be a disk-like self-affine tile generated by an integral expanding matrix $A$ A and a consecutive collinear digit set $\mathcal{D }$ D , and let $f(x)=x^{2}+px+q$ f ( x ) = x 2 + px + q be the characteristic polynomial of $A$ A . In the paper, we identify the boundary $\partial T$ ? T with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,\mathcal{D })$ ( A , D ) to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega $ ω , we find the generalized Hausdorff dimension $\dim _H^{\omega } (\partial T)=2\log \rho (M)/\log |q|$ dim H ω ( ? T ) = 2 log ρ ( M ) / log | q | where $\rho (M)$ ρ ( M ) is the spectral radius of certain contact matrix $M$ M . Especially, when $A$ A is a similarity, we obtain the standard Hausdorff dimension $\dim _H (\partial T)=2\log \rho /\log |q|$ dim H ( ? T ) = 2 log ρ / log | q | where $\rho $ ρ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$ x 3 ? ( | p | ? 1 ) x 2 ? ( | q | ? | p | ) x ? | q | , which is simpler than the known result.     

Algebraic Boundaries of \mathrm{SO}(2)-Orbitopes

Let $X\subset \mathbb{A }^{2r}$ X ? A 2 r be a real curve embedded into an even-dimensional affine space. We characterise when the $r$ r th secant variety to $X$ X is an irreducible component of the algebraic boundary of the convex hull of the real points $X(\mathbb{R })$ X ( R ) of $X$ X . This fact is then applied to $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes and to the so called Barvinok–Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes, we find all irreducible components of their algebraic boundary.     

Improved Enumeration of Simple Topological Graphs

A simple topological graph $T=(V(T), E(T))$ T = ( V ( T ) , E ( T ) ) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs $G$ G and $H$ H are isomorphic if $H$ H can be obtained from $G$ G by a homeomorphism of the sphere, and weakly isomorphic if $G$ G and $H$ H have the same set of pairs of crossing edges. We generalize results of Pach and Tóth and the author’s previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph $G$ G with $n$ n vertices, $m$ m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize $G$ G is at most $2^{O(n^2\log (m/n))}$ 2 O ( n 2 log ( m / n ) ) , and at most $2^{O(mn^{1/2}\log n)}$ 2 O ( m n 1 / 2 log n ) if $m\le n^{3/2}$ m ≤ n 3 / 2 . As a consequence we obtain a new upper bound $2^{O(n^{3/2}\log n)}$ 2 O ( n 3 / 2 log n ) on the number of intersection graphs of $n$ n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with $n$ n vertices to $2^{n^2\cdot \alpha (n)^{O(1)}}$ 2 n 2 · α ( n ) O ( 1 ) , using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize $G$ G is at most $2^{m^2+O(mn)}$ 2 m 2 + O ( m n ) and at least $2^{\Omega (m^2)}$ 2 Ω ( m 2 ) for graphs with $m>(6+\varepsilon )n$ m > ( 6 + ε ) n .     

Many Empty Triangles have a Common Edge

Given a finite point set $X$ X in the plane, the degree of a pair $\{x,y\} \subset X$ { x , y } ? X is the number of empty triangles $t=\mathrm {conv} \{x,y,z\},$ t = conv { x , y , z } , where empty means $t\cap X=\{x,y,z\}.$ t ∩ X = { x , y , z } . Define $\deg X$ deg X as the maximal degree of a pair in $X.$ X . Our main result is that if $X$ X is a random sample of $n$ n independent and uniform points from a fixed convex body, then $\deg X \ge cn/\ln n$ deg X ≥ cn / ln n in expectation.     

Solution of the Propeller Conjecture in \mathbb R ^3

It is shown that every measurable partition $\{A_1,\ldots , A_k\}$ { A 1 , … , A k } of $\mathbb R ^3$ R 3 satisfies 1 $$\begin{aligned} \sum _{i=1}^k\big \Vert \int _{A_i} x\mathrm{{e}}^{-\frac{1}{2}\Vert x\Vert _2^2}\mathrm{{d}}x\big \Vert _2^2\leqslant 9\pi ^2. \end{aligned}$$ ∑ i = 1 k ‖ ∫ A i x e - 1 2 ‖ x ‖ 2 2 d x ‖ 2 2 ? 9 π 2 . Let $\{P_1,P_2,P_3\}$ { P 1 , P 2 , P 3 } be the partition of $\mathbb R ^2$ R 2 into $120^{\circ }$ 120 ° sectors centered at the origin. The bound (1) is sharp, with equality holding if $A_i=P_i\times \mathbb R $ A i = P i × R for $i\in \{1,2,3\}$ i ∈ { 1 , 2 , 3 } and $A_i=\emptyset $ A i = ? for $i\in \{4,\ldots ,k\}$ i ∈ { 4 , … , k } . This settles positively the $3$ 3 -dimensional Propeller Conjecture of Khot and Naor [(Mathematika 55(1-2):129–165, 2009 (FOCS 2008)]. The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the unique games hardness threshold of the kernel clustering problem with $4\times 4$ 4 × 4 centered and spherical hypothesis matrix equals $\frac{2\pi }{3}$ 2 π 3 .     

On Topological Changes in the Delaunay Triangulation of Moving Points

Let $P$ P be a collection of $n$ n points moving along pseudo-algebraic trajectories in the plane. (So, in particular, there are constants $s,c>0$ s , c > 0 such that any four points are co-circular at most $s$ s times, and any three points are collinear at most $c$ c times.) One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a sub-cubic bound, on the maximum number of discrete changes that the Delaunay triangulation ${\mathrm{DT}}(P)$ DT ( P ) of $P$ P experiences during the motion of the points of $P$ P . In this paper, we obtain an upper bound of $O(n^{2+{\varepsilon }})$ O ( n 2 + ε ) , for any ${\varepsilon }>0$ ε > 0 , under the assumptions that (i) any four points can be co-circular at most twice and (ii) either no triple of points can be collinear more than twice or no ordered triple of points can be collinear more than once.     

Nef line bundles on flag bundles on a curve over {\overline{{\mathbb F}}_p}

Let E be a vector bundle of rank r over an irreducible smooth projective curve X defined over the field ${\overline{{\mathbb F}}_p}$ F ¯ p . For fixed integers ${r_1\, , \ldots\, , r_\nu}$ r 1 , ... , r ν with ${1\, \leq\, r_1\, <\, \cdots\, <\, r_\nu\, <\, r}$ 1 ≤ r 1 < ? < r ν < r , let ${\text{Fl}(E)}$ Fl ( E ) be the corresponding flag bundle over X associated to E. Let ${\xi\, \longrightarrow \, {\rm Fl}(E)}$ ξ ? Fl ( E ) be a line bundle such that for every pair of the form ${(C\, ,\phi)}$ ( C , ? ) , where C is an irreducible smooth projective curve defined over ${\overline{\mathbb F}_p}$ F ¯ p and ${\phi\, :\, C\, \longrightarrow\, {\rm Fl}(E)}$ ? : C ? Fl ( E ) is a nonconstant morphism, the inequality ${{\rm degree}(\phi^* \xi)\, > \, 0}$ degree ( ? ? ξ ) > 0 holds. We prove that the line bundle ${\xi}$ ξ is ample.     

Approximating Tverberg Points in Linear Time for Any Fixed Dimension

Let $P \subseteq \mathbb{R }^d$ P ? R d be a $d$ d -dimensional $n$ n -point set. A Tverberg partition is a partition of $P$ P into $r$ r sets $P_1, \dots , P_r$ P 1 , ? , P r such that the convex hulls $\hbox {conv}(P_1), \dots , \hbox {conv}(P_r)$ conv ( P 1 ) , ? , conv ( P r ) have non-empty intersection. A point in $\bigcap _{i=1}^{r} \hbox {conv}(P_i)$ ? i = 1 r conv ( P i ) is called a Tverberg point of depth $r$ r for $P$ P . A classic result by Tverberg shows that there always exists a Tverberg partition of size $\lceil n/(d+1) \rceil $ ? n / ( d + 1 ) ? , but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest. We describe a deterministic algorithm that finds a Tverberg partition of size $\lceil n/4(d+1)^3 \rceil $ ? n / 4 ( d + 1 ) 3 ? in time $d^{O(\log d)} n$ d O ( log d ) n . This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy (Comput Geom Theory Appl 43(8):647–654, 2010).     

*-Quantizations of Fourier–Mukai Transforms

We study deformations of Fourier–Mukai transforms in general complex analytic settings. Suppose X and Y are complex manifolds, and let P be a coherent sheaf on X ×  Y. Suppose that the Fourier–Mukai transform ${\Phi}$ Φ given by the kernel P is an equivalence between the coherent derived categories of X and of Y. Suppose also that we are given a formal *-quantization ${\mathbb{X}}$ X of X. Our main result is that ${\mathbb{X}}$ X gives rise to a unique formal *-quantization ${\mathbb{Y}}$ Y of Y. For the statement to hold, *-quantizations must be understood in the framework of stacks of algebroids. The quantization ${\mathbb{Y}}$ Y is uniquely determined by the condition that ${\Phi}$ Φ deforms to an equivalence between the derived categories of ${\mathbb{X}}$ X and ${\mathbb{Y}}$ Y . Equivalently, the condition is that P deforms to a coherent sheaf ${\tilde{P}}$ P ~ on the formal *-quantization ${\mathbb{X} \times\mathbb{Y}^{op}}$ X × Y o p of X × Y; here ${\mathbb{Y}^{op}}$ Y o p is the opposite of the quantization ${\mathbb{Y}}$ Y .