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1. CMB Online first

Haase, Christian; Hofmann, Jan
 Convex-normal (pairs of) polytopes In 2012 Gubeladze (Adv. Math. 2012) introduced the notion of $k$-convex-normal polytopes to show that integral polytopes all of whose edges are longer than $4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no difference between $k$- and $(k+1)$-convex-normality (for $k\geq 3$) and improve the bound to $2d(d+1)$. In the second part we extend the definition to pairs of polytopes. Given two rational polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement of the normal fan of $Q$. If every edge $e_P$ of $P$ is at least $d$ times as long as the corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap \mathbb{Z}^d = (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$. Keywords:integer decomposition property, integrally closed, projectively normal, lattice polytopesCategories:52B20, 14M25, 90C10

2. CMB Online first

Werner, Elisabeth; Ye, Deping
 Mixed $f$-divergence for multiple pairs of measures In this paper, the concept of the classical $f$-divergence for a pair of measures is extended to the mixed $f$-divergence for multiple pairs of measures. The mixed $f$-divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed $f$-divergence are established, such as permutation invariance and symmetry in distributions. An Alexandrov-Fenchel type inequality and an isoperimetric inequality for the mixed $f$-divergence are proved. Keywords:Alexandrov-Fenchel inequality, $f$-dissimilarity, $f$-divergence, isoperimetric inequalityCategories:28-XX, 52-XX, 60-XX

3. CMB 2015 (vol 59 pp. 204)

Spektor, Susanna
 Restricted Khinchine Inequality We prove a Khintchine type inequality under the assumption that the sum of Rademacher random variables equals zero. We also show a new tail-bound for a hypergeometric random variable. Keywords:Khintchine inequality, Kahane inequality, Rademacher random variables, hypergeometric distribution.Categories:46B06, 60E15, 52A23, 46B09

4. CMB 2014 (vol 58 pp. 561)

Martinez-Maure, Yves
 Plane Lorentzian and Fuchsian Hedgehogs Parts of the Brunn-Minkowski theory can be extended to hedgehogs, which are envelopes of families of affine hyperplanes parametrized by their Gauss map. F. Fillastre introduced Fuchsian convex bodies, which are the closed convex sets of Lorentz-Minkowski space that are globally invariant under the action of a Fuchsian group. In this paper, we undertake a study of plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the Fuchsian analogues of classical geometrical inequalities (analogues which are reversed as compared to classical ones). Keywords:Fuchsian and Lorentzian hedgehogs, evolute, duality, convolution, reversed isoperimetric inequality, reversed Bonnesen inequalityCategories:52A40, 52A55, 53A04, 53B30

5. CMB 2014 (vol 57 pp. 658)

Thang, Nguyen Tat
 Admissibility of Local Systems for some Classes of Line Arrangements Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$ and let $M$ be its complement. A rank one local system $\mathcal{L}$ on $M$ is admissible if roughly speaking the cohomology groups $H^m(M,\mathcal{L})$ can be computed directly from the cohomology algebra $H^{*}(M,\mathbb{C})$. In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$. As a result, we obtain some properties of the characteristic variety $\mathcal{V}_1(M)$ and the Resonance variety $\mathcal{R}_1(M)$. Keywords:admissible local system, line arrangement, characteristic variety, multinet, resonance varietyCategories:14F99, 32S22, 52C35, 05A18, 05C40, 14H50

6. CMB 2013 (vol 57 pp. 640)

 Equilateral Sets and a SchÃ¼tte Theorem for the $4$-norm A well-known theorem of SchÃ¼tte (1963) gives a sharp lower bound for the ratio of the maximum and minimum distances between $n+2$ points in $n$-dimensional Euclidean space. In this note we adapt BÃ¡rÃ¡ny's elegant proof (1994) of this theorem to the space $\ell_4^n$. This gives a new proof that the largest cardinality of an equilateral set in $\ell_4^n$ is $n+1$, and gives a constructive bound for an interval $(4-\varepsilon_n,4+\varepsilon_n)$ of values of $p$ close to $4$ for which it is known that the largest cardinality of an equilateral set in $\ell_p^n$ is $n+1$. Categories:46B20, 52A21, 52C17

7. CMB 2012 (vol 57 pp. 42)

 Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls e prove that, given any covering of any infinite-dimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball. Keywords:point finite coverings, slices, polyhedral spaces, Hilbert spacesCategories:46B20, 46C05, 52C17

8. CMB 2012 (vol 57 pp. 61)

Geschke, Stefan
 2-dimensional Convexity Numbers and $P_4$-free Graphs For $S\subseteq\mathbb R^n$ a set $C\subseteq S$ is an $m$-clique if the convex hull of no $m$-element subset of $C$ is contained in $S$. We show that there is essentially just one way to construct a closed set $S\subseteq\mathbb R^2$ without an uncountable $3$-clique that is not the union of countably many convex sets. In particular, all such sets have the same convexity number; that is, they require the same number of convex subsets to cover them. The main result follows from an analysis of the convex structure of closed sets in $\mathbb R^2$ without uncountable 3-cliques in terms of clopen, $P_4$-free graphs on Polish spaces. Keywords:convex cover, convexity number, continuous coloring, perfect graph, cographCategories:52A10, 03E17, 03E75

9. CMB 2012 (vol 57 pp. 178)

Rabier, Patrick J.
 Quasiconvexity and Density Topology We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then $\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while $\inf_{U}f=\operatorname{ess\,inf}_{U}f$ if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second) property is typical of lsc (usc) functions and, even when $U$ is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions. This property ensures that the pointwise extrema of $f$ on any nonempty density open subset can be arbitrarily closely approximated by values of $f$ achieved on large'' subsets, which may be of relevance in a variety of issues. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero. Keywords:density topology, quasiconvex function, approximate continuity, point of continuityCategories:52A41, 26B05

10. CMB 2012 (vol 57 pp. 3)

Adamczak, Radosław; Latała, Rafał; Litvak, Alexander E.; Oleszkiewicz, Krzysztof; Pajor, Alain; Tomczak-Jaegermann, Nicole
 A Short Proof of Paouris' Inequality We give a short proof of a result of G.~Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\mathbb{R}^n,$ stating that for every $t\geq 1$, $\mathbb{P} \big( |X|\geq ct\sqrt n\big)\leq \exp(-t\sqrt n).$ More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, $(\mathbb{E}|X|^p)^{1/p}\sim \mathbb{E} |X|+\sup_{z\in S^{n-1}}(\mathbb{E} |\langle z,X\rangle|^p)^{1/p}.$ Keywords:log-concave random vectors, deviation inequalitiesCategories:46B06, 46B09, 52A23

11. CMB 2011 (vol 55 pp. 498)

Fradelizi, Matthieu; Paouris, Grigoris; Schütt, Carsten
 Simplices in the Euclidean Ball We establish some inequalities for the second moment $$\frac{1}{|K|} \int_{K}|x|_2^2 \,dx$$ of a convex body $K$ under various assumptions on the position of $K$. Keywords:convex body, simplexCategory:52A20

12. CMB 2011 (vol 55 pp. 697)

Borwein, Jonathan M.; Vanderwerff, Jon
 Constructions of Uniformly Convex Functions We give precise conditions under which the composition of a norm with a convex function yields a uniformly convex function on a Banach space. Various applications are given to functions of power type. The results are dualized to study uniform smoothness and several examples are provided. Keywords:convex function, uniformly convex function, uniformly smooth function, power type, Fenchel conjugate, composition, normCategories:52A41, 46G05, 46N10, 49J50, 90C25

13. CMB 2011 (vol 55 pp. 767)

Martini, Horst; Wu, Senlin
 On Zindler Curves in Normed Planes We extend the notion of Zindler curve from the Euclidean plane to normed planes. A characterization of Zindler curves for general normed planes is given, and the relation between Zindler curves and curves of constant area-halving distances in such planes is discussed. Keywords:rc length, area-halving distance, Birkhoff orthogonality, convex curve, halving pair, halving distance, isosceles orthogonality, midpoint curve, Minkowski plane, normed plane, Zindler curveCategories:52A21, 52A10, 46C15

14. CMB 2011 (vol 55 pp. 487)

Deng, Xinghua; Moody, Robert V.
 Weighted Model Sets and their Higher Point-Correlations Examples of distinct weighted model sets with equal $2,3,4, 5$-point correlations are given. Keywords:model sets, correlations, diffractionCategories:52C23, 51P05, 74E15, 60G55

15. CMB 2011 (vol 55 pp. 98)

Glied, Svenja
 Similarity and Coincidence Isometries for Modules The groups of (linear) similarity and coincidence isometries of certain modules $\varGamma$ in $d$-dimensional Euclidean space, which naturally occur in quasicrystallography, are considered. It is shown that the structure of the factor group of similarity modulo coincidence isometries is the direct sum of cyclic groups of prime power orders that divide $d$. In particular, if the dimension $d$ is a prime number $p$, the factor group is an elementary abelian $p$-group. This generalizes previous results obtained for lattices to situations relevant in quasicrystallography. Categories:20H15, 82D25, 52C23

16. CMB 2011 (vol 54 pp. 726)

Ostrovskii, M. I.
 Auerbach Bases and Minimal Volume Sufficient Enlargements Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have exotic'' minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having exotic'' minimal-volume sufficient enlargements in terms of Auerbach bases. Keywords:Banach space, Auerbach basis, sufficient enlargementCategories:46B07, 52A21, 46B15

17. CMB 2010 (vol 54 pp. 561)

Uren, James J.
 A Note on Toric Varieties Associated with Moduli Spaces In this note we give a brief review of the construction of a toric variety $\mathcal{V}$ coming from a genus $g \geq 2$ Riemann surface $\Sigma^g$ equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L.~C. Jeffrey. A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety $DM_g$, the so-called \emph{Delzant model of moduli space}, for each genus $g.$ We conclude this note with some basic facts about the moment polytopes of the varieties $\mathcal{V}.$ In particular, we show that the varieties $DM_g$ constructed by Tyurin, and claimed to be smooth, are in fact singular for $g \geq 3.$ Categories:14M25, 52B20

18. CMB 2010 (vol 53 pp. 614)

Böröczky, Károly J.; Schneider, Rolf
 The Mean Width of Circumscribed Random Polytopes For a given convex body $K$ in ${\mathbb R}^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of $K^{(n)}$ and $K$ as $n$ tends to infinity. For a simplicial polytope $P$, a precise asymptotic formula for the difference of the mean widths of $P^{(n)}$ and $P$ is obtained. Keywords:random polytope, mean width, approximationCategories:52A22, 60D05, 52A27

19. CMB 2010 (vol 53 pp. 394)

 On Nearly Equilateral Simplices and Nearly lâ Spaces By $\textrm{d}(X,Y)$ we denote the (multiplicative) Banach--Mazur distance between two normed spaces $X$ and $Y.$ Let $X$ be an $n$-dimensional normed space with $\textrm{d}(X,\ell_\infty^n) \le 2,$ where $\ell_\infty^n$ stands for $\mathbb{R}^n$ endowed with the norm $\|(x_1,\dots,x_n)\|_\infty := \max \{|x_1|,\dots, |x_n| \}.$ Then every metric space $(S,\rho)$ of cardinality $n+1$ with norm $\rho$ satisfying the condition $\max D / \min D \le 2/ \textrm{d}(X,\ell_\infty^n)$ for $D:=\{ \rho(a,b) : a, b \in S, \ a \ne b\}$ can be isometrically embedded into $X.$ Categories:52A21, 51F99, 52C99

20. CMB 2009 (vol 53 pp. 3)

 A Combinatorial Reciprocity Theorem for Hyperplane Arrangements Given a nonnegative integer $m$ and a finite collection $\mathcal A$ of linear forms on $\mathcal Q^d$, the arrangement of affine hyperplanes in $\mathcal Q^d$ defined by the equations $\alpha(x) = k$ for $\alpha \in \mathcal A$ and integers $k \in [-m, m]$ is denoted by $\mathcal A^m$. It is proved that the coefficients of the characteristic polynomial of $\mathcal A^m$ are quasi-polynomials in $m$ and that they satisfy a simple combinatorial reciprocity law. Categories:52C35, 05E99

21. CMB 2009 (vol 52 pp. 451)

Pach, János; Tardos, Gábor; Tóth, Géza
 Indecomposable Coverings 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 $\cal D$ 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 $D\in{\cal D}$ all of whose points are of the same color. Categories:52C15, 05C15

22. CMB 2009 (vol 52 pp. 327)

Berman, Leah Wrenn; Bokowski, Jürgen; Grünbaum, Branko; Pisanski, Toma\v{z}
 Geometric Floral'' Configurations 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. Categories:52C30, 52C99

23. CMB 2009 (vol 52 pp. 342)

Bezdek, K.; Kiss, Gy.
 On the X-ray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width 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 $6$. Keywords:almost smooth convex body, convex body of constant width, weakly neighbourly antipodal convex polytope, Illumination Conjecture, X-ray number, X-ray ConjectureCategories:52A20, 52A37, 52C17, 52C35

24. CMB 2009 (vol 52 pp. 349)

Campi, Stefano; Gronchi, Paolo
 On Projection Bodies of Order One The projection body of order one $\Pi_1K$ of a convex body $K$ in $\R^n$ 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 $\Pi_1K$, 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 $n-1$. Category:52A40

25. CMB 2009 (vol 52 pp. 361)

Tóth, Gábor Fejes
 A Note on Covering by Convex Bodies 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 $n\log{n}+n\log\log{n}+5n$. 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 $O(\log{n})$ translates of a lattice arrangement of $K$. Categories:52C07, 52C17
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