Related papers: On reduced polytopes

We study the existence and structure of $d$-polytopes for which the number $f_1$ of edges is small compared to the number $f_0$ of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as…

Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…

Let $P$ be a bounded convex subset of $\mathbb R^n$ of positive volume. Denote the smallest degree of a polynomial $p(X_1,\dots,X_n)$ vanishing on $P\cap\mathbb Z^n$ by $r_P$ and denote the smallest number $u\geq0$ such that every function…

Let $K$ be a convex body in $\Bbb R^{d}$ and $K_{t}$ its floating bodies. There is a polytope with at most $n$ vertices that satisfies $$ K_{t} \subset P_{n} \subset K $$ where $$ n \leq e^{16d} \frac{vol_{d}(K \setminus K_{t})}{t\…

It has recently been established by Below, De Loera, and Richter-Gebert that finding a minimum size (or even just a small) triangulation of a convex polyhedron is NP-complete. Their 3SAT-reduction proof is discussed.

A $\lambda$-convex body in a three-dimensional space form $M^3(c)$ of constant curvature $c$ is a compact convex set $K$ whose boundary $\partial K$ has normal curvatures bounded below by a constant $\lambda>0$ (in a weak sense). Within…

The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in $\mathbb{R}^n$ with smaller volume of all $k$-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to…

Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c$ that to the interior of $\mathcal{P}$. It follows from a lower…

A convex polyhedron, that is, a compact convex subset of $\mathbb{R}^3$ which is the intersection of finitely many closed half-spaces, can be rectified by taking the convex hull of the midpoints of the edges of the polyhedron. We derive…

We consider the following Bezout inequality for mixed volumes: $$V(K_1,\dots,K_r,\Delta[{n-r}])V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(K_i,\Delta[{n-1}])\ \text{ for }2\leq r\leq n.$$ It was shown previously that the inequality is true for…

The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. The paper contains a complete solution to the problem…

We show that learning a convex body in $\RR^d$, given random samples from the body, requires $2^{\Omega(\sqrt{d/\eps})}$ samples. By learning a convex body we mean finding a set having at most $\eps$ relative symmetric difference with the…

We study the problem of covering a given set of $n$ points in a high, $d$-dimensional space by the minimum enclosing polytope of a given arbitrary shape. We present algorithms that work for a large family of shapes, provided either only…

Let $A$ be a polytope in $\mathbb{R}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis…

We consider a quantity that measures the roundness of a bounded, convex $d$-polytope in $\mathbb{R}^d$. We majorise this quantity in terms of the smallest singular value of the matrix of outer unit normals to the facets of the polytope.

Given a polytope $\mathcal{P}$ in $\mathbb{R}^d$ and a subset $U$ of its vertices, is there a triangulation of $\mathcal{P}$ using $d$-simplices that all contain $U$? We answer this question by proving an equivalent and easy-to-check…

A seminal theorem of Tverberg states that any set of $T(r,d)=(r-1)(d+1)+1$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any collection of fewer points in…

The second and fourth authors have conjectured that a certain hollow tetrahedron $\Delta$ of width $2+\sqrt2$ attains the maximum lattice width among all three-dimensional convex bodies. We here prove a local version of this conjecture:…

Recall that a convex body $K$ is in John's position if the unit Euclidean ball is the maximal volume ellipsoid contained in $K$. Approximating convex body in John's position by polytopes we obtain the following results. 1. Let $n>R_n\ge 1$…

Approximating convex bodies is a fundamental question in geometry, which has a wide variety of applications. Given a convex body $K$ in $\textbf{R}^d$ for fixed $d$, the objective is to minimize the number of facets of an approximating…