51. 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 

52. CMB 1999 (vol 42 pp. 380)
53. 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 

54. CMB 1997 (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 

55. CMB 1997 (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 

56. CMB 1997 (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 

57. CMB 1997 (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 
