1. CMB 2016 (vol 60 pp. 350)
 Ma, Yumei

Isometry on Linear $n$Gquasi Normed Spaces
This paper generalizes the Aleksandrov problem: the MazurUlam
theorem on $n$Gquasi normed spaces. It proves that a one$n$distance
preserving mapping is an $n$isometry if and only if it has the
zero$n$Gquasi preserving property, and two kinds of $n$isometries
on $n$Gquasi normed space are equivalent; we generalize the
Benz theorem to nnormed spaces with no restrictions on the dimension
of spaces.
Keywords:$n$Gquasi norm, MazurUlam theorem, Aleksandrov problem, $n$isometry, $n$0distance Categories:46B20, 46B04, 51K05 

2. CMB 2016 (vol 59 pp. 234)
 Beardon, Alan F.

Nondiscrete Frieze Groups
The classification of Euclidean frieze groups into seven conjugacy
classes is well known, and many articles on recreational mathematics
contain frieze patterns that illustrate these classes. However,
it is
only possible to draw these patterns because the subgroup of
translations that leave the pattern invariant is (by definition)
cyclic, and hence discrete. In this paper we classify the conjugacy
classes of frieze groups that contain a nondiscrete subgroup of
translations, and clearly these groups cannot be represented
pictorially in any practical way. In addition, this discussion
sheds
light on why there are only seven conjugacy classes in the classical
case.
Keywords:frieze groups, isometry groups Categories:51M04, 51N30, 20E45 

3. CMB 2014 (vol 57 pp. 390)
 Morita, Jun; Rémy, Bertrand

Simplicity of Some Twin Tree Automorphism Groups with Trivial Commutation Relations
We prove simplicity for incomplete rank 2 KacMoody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs.
We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite).
Nevertheless we use the fact that the latter groups are just infinite
(modulo center).
Keywords:KacMoody group, twin tree, simplicity, root system, building Categories:20G44, 20E42, 51E24 

4. CMB 2011 (vol 55 pp. 487)
5. CMB 2011 (vol 55 pp. 329)
 Kamiya, Shigeyasu; Parker, John R.; Thompson, James M.

NonDiscrete Complex Hyperbolic Triangle Groups of Type $(n,n, \infty;k)$
A complex hyperbolic triangle group is a group
generated by three involutions fixing complex lines in complex
hyperbolic space. Our purpose in this paper is to improve a previous result
and to discuss discreteness of complex hyperbolic
triangle groups of type $(n,n,\infty;k)$.
Keywords:complex hyperbolic triangle group Categories:51M10, 32M15, 53C55, 53C35 

6. CMB 2010 (vol 53 pp. 534)
7. CMB 2010 (vol 53 pp. 394)
 Averkov, Gennadiy

On Nearly Equilateral Simplices and Nearly l_{â} Spaces
By $\textrm{d}(X,Y)$ we denote the (multiplicative) BanachMazur 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 

8. CMB 2009 (vol 52 pp. 435)
 Monson, B.; Schulte, Egon

Modular Reduction in Abstract Polytopes
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 $\mathbb{Z}[\tau]$
(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.
Keywords:abstract polytopes, regular and chiral, Coxeter groups, modular reduction Categories:51M20, 20F55 

9. 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 

10. 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 

11. 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 
