Related papers: On Hadwiger's covering functional for the simplex …

Few-weight codes have been constructed and studied for many years, since their fascinating relations to finite geometries, strongly regular graphs and Boolean functions. Simplex codes are one-weight Griesmer $[\frac{q^k-1}{q-1},k…

The covering number of a finite group $G$, denoted $\sigma(G)$, is the smallest positive integer $k$ such that $G$ is a union of $k$ proper subgroups. We calculate $\sigma(G)$ for a family of primitive groups $G$ with a unique minimal…

Let $(K,\Delta)$ be a Hermitian form field and $n\geq 3$. We prove that if $\sigma\in U_{2n+1}(K,\Delta)$ is a unitary matrix of level $(K,\Delta)$, then any short root transvection $T_{ij}(x)$ is a product of $4$ elementary unitary…

This paper contains a classication of the regular minimal abstract polytopes that act as covers for the convex polyhedral prisms and antiprisms. It includes a detailed discussion of their topological structure, and completes the enumeration…

It is a celebrated result in early combinatorics that, in bipartite graphs, the size of maximum matching is equal to the size of a minimum vertex cover. K\H{o}nig's proof of this fact gave an algorithm for finding a minimum vertex cover…

We prove that for every centrally symmetric convex polygon Q, there exists a constant alpha such that any alpha*k-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound…

If a convex body $K \subset \mathbb{R}^n$ is covered by the union of convex bodies $C_1, \ldots, C_N$, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between…

In this short note, we prove Hadwiger's conjecture for strongly monotypic polytopes.

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far reaching generalization of K\"onig's theorem is only known to be true for $r\leq 3$,…

Let $\Omega$ be a bounded closed convex set in ${\mathbb R}^d$ with non-empty interior, and let ${\cal C}_r(\Omega)$ be the class of convex functions on $\Omega$ with $L^r$-norm bounded by $1$. We obtain sharp estimates of the…

We prove that for any $n\in \mathbb{N}$ there is a convex body $K\subseteq \mathbb{R}^n$ whose surface area is at most $n^{\frac12+o(1)}$, yet the translates of $K$ by the integer lattice $\mathbb{Z}^n$ tile $\mathbb{R}^n$.

We provide an affirmative answer to a problem posed by Barvinok and Veomett, showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of a sub-exponential dimension. Moreover, we…

A small cover was introduced by Davis and Januszkiewicz as an $n$-dimensional closed manifold with a locally standard $Z_2)^n$-action such that its orbit space is a simple convex polytope. There exist a one-to-one correspondence between…

The boundedness of the kissing numbers of convex bodies has been known to Hadwiger for long. We present an application of it to the sum-product estimate…

The generalized circumradius of a set of points $A \subseteq \mathbb{R}^d$ with respect to a convex body $K$ equals the minimum value of $\lambda \geq 0$ such that $A$ is contained in a translate of $\lambda K$. Each choice of $K$ gives a…

New proofs of the Hadwiger theorem for smooth and for continuous valuations on convex functions are obtained, and the Klain-Schneider theorem on convex functions is established. In addition, an extension theorem for valuations defined on…

Given a convex $n$-gon $P$ and a positive integer $m$ such that $3\le m\le n-1$, let $Q$ denote the largest area convex $m$-gon contained in $P$. We are interested in the minimum value of $\Delta(Q)/\Delta(P)$, the ratio of the areas of…

Let $K_q(n,r)$ denote the minimum size of a $q$-ary covering code of word length $n$ and covering radius $r$. In other words, $K_q(n,r)$ is the minimum size of a set of $q$-ary codewords of length $n$ such that the Hamming balls of radius…

A translation structure equips a Riemann surface with a singular flat metric. Not much is known about the shape of a random translation surface. We compute an upper bound on the expected value of the covering radius of a translation surface…

It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let $X$ be a locally convex Hausdorff space and $f \colon X \mapsto \left( - \infty , \infty \right]$ a function with compact…