26. 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 

27. 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 axisparallel 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 

28. 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 

29. 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 

30. CMB 2006 (vol 49 pp. 536)
31. 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 

32. 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 

33. 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 

34. 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 

35. CMB 2005 (vol 48 pp. 302)
36. 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 

37. 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 

38. 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 

39. 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 

40. 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 

41. CMB 2002 (vol 45 pp. 537)
42. 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 

43. 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 

44. 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 

45. CMB 2002 (vol 45 pp. 232)
 Ji, Min; Shen, Zhongmin

On Strongly Convex Indicatrices in Minkowski Geometry
The geometry of indicatrices is the foundation of Minkowski geometry.
A strongly convex indicatrix in a vector space is a strongly convex
hypersurface. It admits a Riemannian metric and has a distinguished
invariant(Cartan) torsion. We prove the existence of nontrivial
strongly convex indicatrices with vanishing mean torsion and discuss
the relationship between the mean torsion and the Riemannian curvature
tensor for indicatrices of Randers type.
Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35 

46. CMB 2002 (vol 45 pp. 123)
 Moody, Robert V.

Uniform Distribution in Model Sets
We give a new measuretheoretical proof of the uniform distribution
property of points in model sets (cut and project sets). Each model
set comes as a member of a family of related model sets, obtained by
joint translation in its ambient (the `physical') space and its
internal space. We prove, assuming only that the window defining the
model set is measurable with compact closure, that almost surely the
distribution of points in any model set from such a family is uniform
in the sense of Weyl, and almost surely the model set is pure point
diffractive.
Categories:52C23, 11K70, 28D05, 37A30 

47. CMB 2000 (vol 43 pp. 427)
 Ivey, Thomas A.

Helices, Hasimoto Surfaces and BÃ¤cklund Transformations
Travelling wave solutions to the vortex filament flow generated by
elastica produce surfaces in $\R^3$ that carry mutually orthogonal
foliations by geodesics and by helices. These surfaces are classified
in the special cases where the helices are all congruent or are all
generated by a single screw motion. The first case yields a new
characterization for the B\"acklund transformation for constant
torsion curves in $\R^3$, previously derived from the wellknown
transformation for pseudospherical surfaces. A similar investigation
for surfaces in $H^3$ or $S^3$ leads to a new transformation for
constant torsion curves in those spaces that is also derived from
pseudospherical surfaces.
Keywords:surfaces, filament flow, BÃ¤cklund transformations Categories:53A05, 58F37, 52C42, 58A15 

48. CMB 2000 (vol 43 pp. 368)
 Litvak, A. E.

KahaneKhinchin's Inequality for QuasiNorms
We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about
equivalence of $L_q$norms of logconcave random variables
(KahaneKhinchin's inequality) to the quasiconvex case. We
construct examples of quasiconvex bodies $K_n \subset \R$ which
demonstrate that this equivalence fails for uniformly distributed
vector on $K_n$ (recall that the uniformly distributed vector on a
convex body is logconcave). Our examples also show the lack of the
exponential decay of the ``tail" volume (for convex bodies such
decay was proved by M.~Gromov and V.~Milman).
Categories:46B09, 52A30, 60B11 

49. CMB 1999 (vol 42 pp. 380)
50. CMB 1999 (vol 42 pp. 237)
 Thompson, A. C.

On Benson's Definition of Area in Minkowski Space
Let $(X, \norm)$ be a Minkowski space (finite dimensional Banach
space) with unit ball $B$. Various definitions of surface area are
possible in $X$. Here we explore the one given by Benson
\cite{ben1}, \cite{ben2}. In particular, we show that this
definition is convex and give details about the nature of the
solution to the isoperimetric problem.
Categories:52A21, 52A38 
