Related papers: A Proof of the Simplex Mean Width Conjecture
Answering Tarski's plank problem, Bang showed in 1951 that it is impossible to cover a convex body $K \subset \mathbb{R}^d$ with $d \geq 1$ by planks whose total width is less than the minimal width $w(K)$ of $K$. In 2003, A. Bezdek asked…
According to the Erd\H{o}s-Szekeres theorem, for every $n$, a sufficiently large set of points in general position in the plane contains $n$ in convex position. In this note we investigate the line version of this result, that is, we want…
In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the $n$-dimensional volume of a regular simplex and the $(n-1)$-dimensional volume of its facets. In addition to the…
Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume for a general convex body that applies in any space dimension, including both the rotationally-averaged…
We study the dual variants of the Erd\H{o}s's distinct distances and unit distance problems. Instead of considering distances determined by points, we consider simplex volumes determined by hyperplanes. We investigate: (1) the maximum…
K. Bezdek and Gy. Kiss showed that existence of origin-symmetric coverings of unit sphere in $\mathbb{E}^n$ by at most $2^n$ congruent spherical caps with radius not exceeding $\arccos\sqrt{\frac{n-1}{2n}}$ implies the $X$-ray conjecture…
The present work considers the properties of classes of generally convex sets in the plane known as $1$-semiconvex and weakly $1$-semiconvex. More specifically, the examples of open and closed weakly $1$-semiconvex but non $1$-semiconvex…
This paper proves the formula \nu(d) =1 for d=1 and \nu(d) = max(1/4 +1/d, 1/2) for d > 1 for the root mean square displacement exponent \nu(d) of the self-avoiding walk (SAW) in Z^d, and thus, resolves some major long-standing open…
A finite set of unlabelled points in Euclidean space is the simplest representation of many real objects from mineral rocks to sculptures. Since most solid objects are rigid, their natural equivalence is rigid motion or isometry maintaining…
The Flatness theorem states that the maximum lattice width ${\rm Flt}(d)$ of a $d$-dimensional lattice-free convex set is finite. It is the key ingredient for Lenstra's algorithm for integer programming in fixed dimension, and much work has…
Among integral polytopes (vertices with integral coordinates), lattice-free polytopes - intersecting the lattice ONLY at their vertices- are of particular interestin combinatorics and geometry of numbers. A natural question is to measure…
Our study focuses on determining the best weight windows for a weighted moving average smoother under squared loss. We show that there exists an optimal weight window that is symmetrical around its center. We study the class of tapered…
The Blaschke-Leichtweiss theorem (Abh. Math. Sem. Univ. Hamburg 75: 257-284, 2005) states that the smallest area convex domain of constant width $w$ in the $2$-dimensional spherical space ${\mathbb S}^2$ is the spherical Reuleaux triangle…
The longstanding Godbersen's conjecture states that for any convex body $K \subset \mathbb R^n$ of volume $1$ and any $j \in \{0, \ldots, n\}$, the mixed volume $V_j = V(K[j], -K[n - j])$ is bounded by $\binom{n}{j}$, with equality if and…
Let $n\geq 3$, and let $B_1^n$ be the standard $n$-dimensional cross-polytope (i.e. the convex hull of standard coordinate vectors and their negatives). We show that there exists a symmetric convex body $\mathcal G_m$ in ${\mathbb R}^n$…
We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in John's position, as well as for their polar bodies. These estimates extend some well-known results for convex…
A packing of translates of a convex body in the $d$-dimensional Euclidean space $\mathbb{E}^d$ is said to be totally separable if any two packing elements can be separated by a hyperplane of $\mathbb{E}^{d}$ disjoint from the interior of…
We formulate several conjectures on mean convex domains in the Euclidean spaces, as well as in more general spaces with lower bonds on their scalar curvatures, and prove a few theorems motivating these conjectures.
In a convex mosaic in $\mathbb{R} ^d$ we denote the average number of vertices of a cell by $\bar v$ and the average number of cells meeting at a node by $\bar n$. Except for the $d=2$ planar case, there is no known formula prohibiting…
A {\em slab} (or plank) of width $w$ is a part of the $d$-dimensional space that lies between two parallel hyperplanes at distance $w$ from each other. It is conjectured that any slabs $S_1, S_2,\ldots$ whose total width is divergent have…