It is with great pleasure that we present this special geometry issue of the Canadian Mathematical Bulletin dedicated to Professor Tibor Bisztriczky on occasion of his sixtieth birthday. All articles were prepared by the contributors especially for this collection. This issue presents new results from current investigations in the fields of the combinatorial theory of polytopes, convexity, the theory of packing and covering, and combinatorial geometry, subjects where Professor Bisztriczky made significant contributions and where he is currently engaged in active research. Here we take the opportunity to thank the contributors for submitting their most valuable work and the referees who devoted their efforts and time to make this special volume possible.

With an increase in size, configurations of points and lines in the plane usually become complicated and hard to analyze. The “floral” configurations we are introducing here represent a new type that makes accessible and visually intelligible even configurations of considerable size. This is achieved by combining a large degree of symmetry with a hierarchical construction. Depending on the details of the interdependence of these aspects, there are several subtypes that are described and investigated.

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.

The projection body of order one ${{\Pi }_{1}}K$ of a convex body $K$ in ${{\mathbb{R}}^{n}}$ is the body whose support function is, up to a constant, the average mean width of the orthogonal projections of $K$ onto hyperplanes through the origin.

The paper contains an inequality for the support function of ${{\Pi }_{1}}K$ , which implies in particular that such a function is strictly convex, unless $K$ has dimension one or two. Furthermore, an existence problem related to the reconstruction of a convex body is discussed to highlight the different behavior of the area measures of order one and of order $n\,-\,1$ .

A classical theorem of Rogers states that for any convex body $K$ in $n$ -dimensional Euclidean space there exists a covering of the space by translates of $K$ with density not exceeding $n\,\log \,n\,+\,n\,\log \,\log \,n\,+\,5$ . Rogers’ theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of $n$ the same bound can be attained by a covering which is the union of $O\left( \log \,n \right)$ translates of a lattice arrangement of $K$ .

We construct, for all $d\,\ge \,4$ , a cellulation of ${{\mathbb{S}}^{d-1}}$ . We prove that these cellulations cannot be polytopal with maximal combinatorial symmetry. Such non-realizability phenomenon was first described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and, to the knowledge of the author, until now there have not been any known examples in higher dimensions. As a starting point for the construction, we introduce a new class of (Wythoffian) uniform polytopes, which we call duplexes. In proving our main result, we use some tools that we developed earlier while studying perfect polytopes. In particular, we prove perfectness of the duplexes; furthermore, we prove and make use of the perfectness of another new class of polytopes which we obtain by a variant of the so-called $E$ -construction introduced by Eppstein, Kuperberg and Ziegler.

In this note we present inequalities relating the successive minima of an $o$ -symmetric convex body and the successive inner and outer radii of the body. These inequalities join known inequalities involving only either the successive minima or the successive radii.

Let $K$ denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of $K$ is said to have property $T\left( k \right)$ if for every subset of at most $k$ translates there exists a common line transversal intersecting all of them. The integer $k$ is the stabbing level of the family. Two translates ${{K}_{i}}\,=\,K\,+\,{{c}_{i}}$ and ${{K}_{j}}\,=\,K\,+\,{{c}_{j}}$ are said to be $\sigma$ -disjoint if $\sigma K\,+\,{{c}_{i}}$ and $\sigma K\,+\,{{c}_{j}}$ are disjoint. A recent Helly-type result claims that for every $\sigma \,>\,0$ there exists an integer $k\left( \sigma\right)$ such that if a family of $\sigma$ -disjoint unit diameter discs has property $T\left( k \right)|k\ge k\left( \sigma\right)$ , then there exists a straight line meeting all members of the family. In the first part of the paper we give the extension of this theorem to translates of an oval $k$ . The asymptotic behavior of $k\left( \sigma\right)$ for $\sigma \,\to \,0$ is considered as well.

Katchalski and Lewis proved the existence of a constant $r$ such that for every pairwise disjoint family of translates of an oval $K$ with property $T\left( 3 \right)$ a straight line can be found meeting all but at most $r$ members of the family. In the second part of the paper $\sigma$ -disjoint families of translates of $K$ are considered and the relation of $\sigma$ and the residue $r$ is investigated. The asymptotic behavior of $r\left( \sigma\right)$ for $\sigma \,\to \,0$ is also discussed.

We shall prove the following shaken Rogers's theorem for homothetic sections: Let $K$ and $L$ be strictly convex bodies and suppose that for every plane $H$ through the origin we can choose continuously sections of $K$ and $L$ , parallel to $H$ , which are directly homothetic. Then $K$ and $L$ are directly homothetic.

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ${{\mathbb{R}}^{d}}$ is ${{2}^{d}}$ . Naszódi proved that the quantity in question is not larger than ${{2}^{d+1}}$ . We present an improvement to this result by proving the upper bound $3\,\cdot \,{{2}^{d-1}}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

A Halin graph is a graph $H\,=\,T\,\cup \,C$ , where $T$ is a tree with no vertex of degree two, and $C$ is a cycle connecting the end-vertices of $T$ in the cyclic order determined by a plane embedding of $T$ . In this paper, we define classes of generalized Halin graphs, called $k$ -Halin graphs, and investigate their Hamiltonian properties.

We investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter by $k$ unit circles. In particular, we study the cases $k\,=\,3$ , $k\,=\,4$ , and $k\,=\,7$ . For $k\,=\,3$ and $k\,=\,4$ , the diameters under consideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essential role in our proofs, namely Minkowskian bisectors, $d$ -segments, and the monotonicity lemma.

The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind: first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in $\mathbb{Z}\left[ \tau\right]$ (with $\tau$ the golden ratio), to construct new regular 4-polytopes of hyperbolic types $\{3,\,5,\,3\}$ and $\{5,\,3,\,5\}$ with automorphism groups given by finite orthogonal groups.

We prove that for every $k\,>\,1$ , there exist $k$ -fold coverings of the plane (i) with strips, (ii) with axis-parallel rectangles, and (iii) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two coverings. We also construct for every $k\,>\,1$ a set of points $P$ and a family of disks $D$ in the plane, each containing at least $k$ elements of $P$ , such that, no matter how we color the points of $P$ with two colors, there exists a disk $D\,\in \,D$ all of whose points are of the same color.

Let $K\,\subset \,{{\mathbb{R}}^{n+1}}$ be a convex body of class ${{C}^{2}}$ with everywhere positive Gauss curvature. We show that there exists a positive number $\delta \left( K \right)$ such that for any $\delta \,\in \,\left( 0,\,\delta \left( K \right) \right)$ we have $\text{Vol}\left( {{K}_{\delta }} \right)\,\cdot \,\text{Vol}\left( {{\left( {{K}_{\delta }} \right)}^{*}} \right)\,\ge \,\text{Vol}\left( K \right)\,\cdot \,\text{Vol}\left( {{K}^{*}} \right)\,\ge \,\text{Vol}\left( {{K}^{\delta }} \right)\,\cdot \,\text{Vol}\left( {{\left( {{K}^{\delta }} \right)}^{*}} \right)$ , where ${{K}_{\delta }}$ , ${{K}^{\delta }}$ and ${{K}^{*}}$ stand for the convex floating body, the illumination body, and the polar of $K$ , respectively. We derive a few consequences of these inequalities.