Related papers: Big convex polytopes or rich hyperplanes

We give improved lower bounds on the minimum number of $k$-holes (empty convex $k$-gons) in a set of $n$ points in general position in the plane, for $k=5,6$.

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according…

We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an…

Let $(P,E)$ be a $(d+1)$-uniform geometric hypergraph, where $P$ is an $n$-point set in general position in $\mathbb{R}^d$ and $E\subseteq {P\choose d+1}$ is a collection of $\epsilon{n\choose d+1}$ $d$-dimensional simplices with vertices…

Let $X$ be a configuration of $n$ points in $\mathbb{R}^d$. What is the maximum number of vertices that $conv(T(X))$ can have among all the possible permissible projective transformations $T$? In this paper, we investigate this and…

Let $f_i(P)$ denote the number of $i$-dimensional faces of a convex polytope $P$. Furthermore, let $S(n,d)$ and $C(n,d)$ denote, respectively, the stacked and the cyclic $d$-dimensional polytopes on $n$ vertices. Our main result is that for…

Let $A,B \subseteq \mathbb{R}^d $ both span $\mathbb{R}^d$ such that $\langle a, b \rangle \in \{0,1\}$ holds for all $a \in A$, $b \in B$. We show that $ |A| \cdot |B| \le (d+1) 2^d $. This allows us to settle a conjecture by Bohn, Faenza,…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular…

In every dimension $d\ge1$, we establish the existence of a constant $v_d>0$ and of a subset $\mathcal U_d$ of $\mathbb R^d$ such that the following holds: $\mathcal C+\mathcal U_d=\mathbb R^d$ for every convex set $\mathcal C\subset…

We improve our earlier upper bound on the numbers of antipodal pairs of points among $n$ points in ${\mathbb{R}}^3$, to $2n^2/5+O(n^c)$, for some $c<2$. We prove that the minimal number of antipodal pairs among $n$ points in convex position…

The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central…

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of $n$ real numbers (for short, \emph{grid}). First, we prove that every such grid contains $\Omega(\log n)$ points in convex…

We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope…

The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed…

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an…

In this note we show that the maximum number of vertices in any polyhedron $P=\{x\in \mathbb{R}^d : Ax\leq b\}$ with $0,1$-constraint matrix $A$ and a real vector $b$ is at most $d!$.

In this paper we study some Erdos type problems in discrete geometry. Our main result is that we show that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $…

We enumerate and classify all stationary logarithmic configurations of d+2 points on the unit (d-1)-sphere in d-dimensions. In particular, we show that the logarithmic energy attains its relative minima at configurations that consist of two…

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of…