26. CMB 2009 (vol 52 pp. 366)
 Gévay, Gábor

A Class of Cellulated Spheres with NonPolytopal Symmetries
We construct, for all $d\geq 4$, a cellulation of $\mathbb S^{d1}$.
We prove that these cellulations cannot be polytopal with maximal
combinatorial symmetry. Such nonrealizability 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 socalled
$E$construction introduced by Eppstein, Kuperberg and Ziegler.
Keywords:CW sphere, polytopality, automorphism group, symmetry group, uniform polytope Categories:52B11, 52B15, 52B70 

27. CMB 2009 (vol 52 pp. 380)
 Henk, Martin; Cifre, Mar\'\i a A. Hernández

Successive Minima and Radii
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.
Keywords:successive minima, inner and outer radii Categories:52A20, 52C07, 52A40, 52A39 

28. CMB 2009 (vol 52 pp. 403)
 JerónimoCastro, J.; Montejano, L.; MoralesAmaya, E.

Shaken Rogers's Theorem for Homothetic Sections
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.
Keywords:convex bodies, homothetic bodies, sections and projections, Rogers's Theorem Category:52A15 

29. CMB 2009 (vol 52 pp. 407)
 Lángi, Zsolt; Naszódi, Márton

On the BezdekPach Conjecture for Centrally Symmetric Convex Bodies
The BezdekPach conjecture asserts that the maximum number of
pairwise touching positive homothetic copies of a convex body in
$\Re^d$ is $2^d$. Nasz\'odi 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 $3\cdot2^{d1}$ for centrally symmetric
bodies. Bezdek and Brass introduced the onesided 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.
Keywords:BezdekPach Conjecture, homothets, packing, Hadwiger number, antipodality Categories:52C17, 51N20, 51K05, 52A21, 52A37 

30. CMB 2009 (vol 52 pp. 424)
 Martini, Horst; Spirova, Margarita

Covering Discs in Minkowski Planes
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 sidelengths 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.
Keywords:affine regular polygon, bisector, circle covering problem, circumradius, $d$segment, Minkowski plane, (strictly convex) normed plane Categories:46B20, 52A21, 52C15 

31. CMB 2009 (vol 52 pp. 464)
 Stancu, Alina

Two Volume Product Inequalities and Their Applications
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 $\delta (K)$ such that for any $\delta \in (0,
\delta(K))$ we have $\Volu(K_{\delta})\cdot
\Volu((K_{\delta})^{\sstar}) \geq \Volu(K)\cdot \Volu(K^{\sstar}) \geq
\Volu(K^{\delta})\cdot \Volu((K^{\delta})^{\sstar})$, where $K_{\delta}$,
$K^{\delta}$ and $K^{\sstar}$ stand for the convex floating body, the
illumination body, and the polar of $K$, respectively. We derive a
few consequences of these inequalities.
Keywords:affine invariants, convex floating bodies, illumination bodies Categories:52A40, 52A38, 52A20 

32. CMB 2009 (vol 40 pp. 158)
 Coxeter, H. S. M.

The trigonometry of hyperbolic tessellations
For positive integers $p$ and $q$ with $(p2)(q2) >
4$ there is, in the hyperbolic plane, a group $[p,q]$
generated by reflections in the three sides of a triangle
$ABC$ with angles $\pi /p$, $\pi/q$, $\pi/2$. Hyperbolic
trigonometry shows that the side $AC$ has length $\psi$,
where $\cosh \psi = c/s$, $c = \cos \pi/q$, $s = \sin\pi/p$.
For a conformal drawing inside the unit circle with centre
$A$, we may take the sides $AB$ and $AC$ to run straight
along radii while $BC$ appears as an arc of a circle
orthogonal to the unit circle. The circle containing this
arc is found to have radius $1/\sinh \psi = s/z$, where $z
= \sqrt{c^2s^2}$, while its centre is at distance $1/\tanh
\psi = c/z$ from $A$. In the hyperbolic triangle $ABC$,
the altitude from $AB$ to the rightangled vertex $C$ is
$\zeta$, where $\sinh\zeta = z$.
Categories:51F15, 51N30, 52A55 

33. CMB 2009 (vol 40 pp. 471)
 Lawrence, Jim

A short proof of Euler's relation for convex polytopes*
The purposen of this paper is to present a short, selfcontained
proof of Euler's relation. The ingredients of this proof are (i) the
principle of inclusion and exclusion of combinatorics and (ii) the
Euler characteristic; a development of the Euler characteristic is included.
Category:52A25 

34. CMB 2009 (vol 40 pp. 356)
 Mazet, Pierre

Principe du maximum et lemme de Schwarz, a valeurs vectorielles
Nous {\'e}tablissons un
th{\'e}or{\`e}me pour les fonctions holomorphes {\`a} valeurs dans une
partie convexe ferm{\'e}e. Ce th{\'e}or{\`e}me pr{\'e}cise
la position des coefficients de Taylor de telles fonctions et peut
{\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des
in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce
th{\'e}or{\`e}me permet de retrouver des versions connues du principe
du maximum et d'obtenir de nouveaux r{\'e}sultats sur les
applications holomorphes {\`a} valeurs vectorielles.
Keywords:Principe du maximum, lemme de Schwarz, points extr{Ã©maux. Categories:30C80, 32A30, 46G20, 52A07 

35. CMB 2009 (vol 40 pp. 10)
 Borwein, Jon; Vanderwerff, Jon

Convex functions on Banach spaces not containing $\ell_1$
There is a sizeable class of results precisely
relating boundedness, convergence and differentiability properties
of continuous convex functions on Banach spaces to whether or
not the space contains an isomorphic copy of $\ell_1$. In this
note, we provide constructions showing that the main such
results do not extend to natural broader classes of functions.
Categories:46A55, 46B20, 52A41 

36. CMB 2007 (vol 50 pp. 474)
 Zhou, Jiazu

On Willmore's Inequality for Submanifolds
Let $M$ be an $m$ dimensional submanifold in the Euclidean space
${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain
some low geometric estimates of the total square mean curvature
$\int_M H^2 d\sigma$. The low bounds are geometric invariants
involving the volume of $M$, the total scalar curvature of $M$,
the Euler characteristic and the circumscribed ball of $M$.
Keywords:submanifold, mean curvature, kinematic formul, scalar curvature Categories:52A22, 53C65, 51C16 

37. CMB 2006 (vol 49 pp. 536)
38. CMB 2006 (vol 49 pp. 185)
 Averkov, Gennadiy

On the Inequality for Volume and Minkowskian Thickness
Given a centrally symmetric convex body $B$ in $\E^d,$ we denote
by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional
Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex
body in $\M^d(B).$ The relationship between volume $V(K)$ and the
Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can
naturally be given by the sharp geometric inequality $V(K) \ge
\alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple
corollary of the RogersShephard inequality we obtain that
$\binom{2d}{d}{}^{1} \le \alpha(B)/V(B) \le 2^{d}$ with equality
on the left attained if and only if $B$ is the difference body of
a simplex and on the right if $B$ is a crosspolytope. The main
result of this paper is that for $d=2$ the equality on the right
implies that $B$ is a parallelogram. The obtained results yield
the sharp upper bound for the modified BanachMazur distance to the
regular hexagon.
Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, BanachMazur compactum, (modified) BanachMazur distance, volume ratio Categories:52A40, 46B20 

39. CMB 2006 (vol 49 pp. 161)
 Agapito, José

Weighted BrianchonGram Decomposition
We give in this note a weighted version of Brianchon and Gram's
decomposition for a simple polytope. We can derive from this
decomposition the weighted polar formula of Agapito and a weighted
version of Brion's theorem, in a manner similar to Haase, where the
unweighted case is worked out. This weighted version of Brianchon
and Gram' decomposition
is a direct consequence of the ordinary BrianchonGram formula.
Category:52B 

40. CMB 2005 (vol 48 pp. 523)
 Düvelmeyer, Nico

Angle Measures and Bisectors in Minkowski Planes
\begin{abstract}
We prove that a Minkowski plane is Euclidean if and only if Busemann's or
Glogovskij's definitions
of angular bisectors coincide
with a bisector defined by an angular measure in the sense of Brass.
In addition, bisectors defined by the area measure coincide with bisectors
defined by the circumference (arc length) measure
if and only if the unit circle is an
equiframed curve.
Keywords:Radon curves, Minkowski geometry, Minkowski planes,, angular bisector, angular measure, equiframed curves Categories:52A10, 52A21 

41. CMB 2005 (vol 48 pp. 414)
 Kaveh, Kiumars

Vector Fields and the Cohomology Ring of Toric Varieties
Let $X$ be a smooth complex
projective variety with a holomorphic vector field with isolated
zero set $Z$. From the results of Carrell and Lieberman
there exists a filtration
$F_0 \subset F_1 \subset \cdots$ of $A(Z)$, the ring of
$\c$valued functions on $Z$, such that $\Gr A(Z) \cong H^*(X,
\c)$ as graded algebras. In this note, for a smooth projective
toric variety and a vector field generated by the action of a
$1$parameter subgroup of the torus, we work out this filtration.
Our main result is an explicit connection between this filtration
and the polytope algebra of $X$.
Keywords:Toric variety, torus action, cohomology ring, simple polytope,, polytope algebra Categories:14M25, 52B20 

42. CMB 2005 (vol 48 pp. 302)
43. CMB 2004 (vol 47 pp. 481)
 Bekjan, Turdebek N.

A New Characterization of Hardy Martingale Cotype Space
We give a new characterization of Hardy martingale cotype
property of complex quasiBanach space by using the existence of a
kind of plurisubharmonic functions. We also characterize the best
constants of Hardy martingale inequalities with values
in the complex quasiBanach space.
Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic function Categories:46B20, 52A07, 60G44 

44. CMB 2004 (vol 47 pp. 246)
 Makai, Endre; Martini, Horst

On Maximal $k$Sections and Related Common Transversals of Convex Bodies
Generalizing results from [MM1] referring
to the intersection body $IK$ and
the crosssection body $CK$ of a convex body
$K \subset \sR^d, \, d \ge 2$,
we prove theorems about maximal $k$sections of convex bodies,
$k \in \{1, \dots, d1\}$,
and, simultaneously, statements
about common maximal
$(d1)$ and $1$transversals of families
of convex bodies.
Categories:52A20, 55Mxx 

45. CMB 2004 (vol 47 pp. 168)
 Baake, Michael; Sing, Bernd

Kolakoski$(3,1)$ Is a (Deformed) Model Set
Unlike the (classical) Kolakoski sequence on the alphabet $\{1,2\}$, its analogue
on $\{1,3\}$ can be related to a primitive substitution rule. Using this connection,
we prove that the corresponding biinfinite fixed point is a regular generic model set
and thus has a pure point diffraction spectrum. The Kolakoski$(3,1)$ sequence is
then obtained as a deformation, without losing the pure point diffraction property.
Categories:52C23, 37B10, 28A80, 43A25 

46. CMB 2003 (vol 46 pp. 373)
 Laugesen, Richard S.; Pritsker, Igor E.

Potential Theory of the FarthestPoint Distance Function
We study the farthestpoint distance function, which measures the
distance from $z \in \mathbb{C}$ to the farthest point or points of
a given compact set $E$ in the plane.
The logarithm of this distance is subharmonic as a function of $z$,
and equals the logarithmic potential of a unique probability measure
with unbounded support. This measure $\sigma_E$ has many interesting
properties that reflect the topology and geometry of the compact set
$E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a
circle, with equality if and only if $E$ is a regular $n$gon for some
odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of
constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$.
Keywords:distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width Categories:31A05, 52A10, 52A40 

47. CMB 2003 (vol 46 pp. 242)
 Litvak, A. E.; Milman, V. D.

Euclidean Sections of Direct Sums of Normed Spaces
We study the dimension of ``random'' Euclidean sections of direct sums of
normed spaces. We compare the obtained results with results from \cite{LMS},
to show that for the direct sums the standard randomness with respect to the
Haar measure on Grassmanian coincides with a much ``weaker'' randomness of
``diagonal'' subspaces (Corollary~\ref{sle} and explanation after). We also
add some relative information on ``phase transition''.
Keywords:Dvoretzky theorem, ``random'' Euclidean section, phase transition in asymptotic convexity Categories:46B07, 46B09, 46B20, 52A21 

48. CMB 2002 (vol 45 pp. 483)
 Baake, Michael

Diffraction of Weighted Lattice Subsets
A Dirac comb of point measures in Euclidean space with bounded
complex weights that is supported on a lattice $\varGamma$ inherits
certain general properties from the lattice structure. In
particular, its autocorrelation admits a factorization into a
continuous function and the uniform lattice Dirac comb, and its
diffraction measure is periodic, with the dual lattice
$\varGamma^*$ as lattice of periods. This statement remains true
in the setting of a locally compact Abelian group whose topology
has a countable base.
Keywords:diffraction, Dirac combs, lattice subsets, homometric sets Categories:52C07, 43A25, 52C23, 43A05 

49. CMB 2002 (vol 45 pp. 697)
 Sirvent, V. F.; Solomyak, B.

Pure Discrete Spectrum for Onedimensional Substitution Systems of Pisot Type
We consider two dynamical systems associated with a substitution of
Pisot type: the usual $\mathbb{Z}$action on a sequence space, and
the $\mathbb{R}$action, which can be defined as a tiling dynamical
system or as a suspension flow. We describe procedures for checking
when these systems have pure discrete spectrum (the ``balanced
pairs algorithm'' and the ``overlap algorithm'') and study the
relation between them. In particular, we show that pure discrete
spectrum for the $\mathbb{R}$action implies pure discrete spectrum
for the $\mathbb{Z}$action, and obtain a partial result in the
other direction. As a corollary, we prove pure discrete spectrum
for every $\mathbb{R}$action associated with a twosymbol
substitution of Pisot type (this is conjectured for an arbitrary
number of symbols).
Categories:37A30, 52C23, 37B10 

50. CMB 2002 (vol 45 pp. 634)
 Lagarias, Jeffrey C.; Pleasants, Peter A. B.

Local Complexity of Delone Sets and Crystallinity
This paper characterizes when a Delone set $X$ in $\mathbb{R}^n$ is an
ideal crystal in terms of restrictions on the number of its local
patches of a given size or on the heterogeneity of their distribution.
For a Delone set $X$, let $N_X (T)$ count the number of
translationinequivalent patches of radius $T$ in $X$ and let
$M_X(T)$ be the minimum radius such that every closed ball of radius
$M_X(T)$ contains the center of a patch of every one of these kinds.
We show that for each of these functions there is a
``gap in the spectrum'' of possible growth rates between being
bounded and having linear growth, and that having sufficiently
slow linear growth is equivalent to $X$ being an ideal crystal.
Explicitly, for $N_X(T)$, if $R$ is the covering radius of $X$
then either $N_X(T)$ is bounded or $N_X (T) \ge T/2R$ for all $T>0$.
The constant $1/2R$ in this bound is best possible in all dimensions.
For $M_X(T)$, either $M_X(T)$ is bounded or $M_X(T)\ge T/3$ for all $T>0$.
Examples show that the constant $1/3$ in this bound cannot be replaced by
any number exceeding $1/2$. We also show that every aperiodic Delone
set $X$ has $M_X(T)\ge c(n) T$ for all $T>0$, for a certain constant $c(n)$
which depends on the dimension $n$ of $X$ and is $>1/3$ when $n>1$.
Keywords:aperiodic set, Delone set, packingcovering constant, sphere packing Categories:52C23, 52C17 
