26. CJM 2004 (vol 56 pp. 472)
 Fonf, Vladimir P.; Veselý, Libor

InfiniteDimensional Polyhedrality
This paper deals with generalizations of the notion of a polytope to infinite
dimensions. The most general definition is the following: a bounded closed
convex subset of a Banach space is called a \emph{polytope} if each of its
finitedimensional affine sections is a (standard) polytope.
We study the relationships between eight known definitions
of infinitedimensional
polyhedrality. We provide a complete isometric
classification of them, which gives
solutions to several open problems.
An almost complete isomorphic classification
is given as well (only one implication remains open).
Categories:46B20, 46B03, 46B04, 52B99 

27. CJM 2004 (vol 56 pp. 529)
 MartínezFinkelshtein, A.; Maymeskul, V.; Rakhmanov, E. A.; Saff, E. B.

Asymptotics for Minimal Discrete Riesz Energy on Curves in $\R^d$
We consider the $s$energy
$$
E(\ZZ_n;s)=\sum_{i \neq j} K(\z_{i,n}z_{j,n}\;s)
$$
for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets
$\Ga$ in $\R^d$ having finite onedimensional Hausdorff measure, where
$$
K(t;s)=
\begin{cases}
t^{s} ,& \mbox{if } s>0, \\
\ln t, & \mbox{if } s=0,
\end{cases}
$$
is the Riesz kernel. Asymptotics for the minimum $s$energy and the
distribution of minimizing sequences of points is studied. In
particular, we prove that, for $s\geq 1$, the minimizing nodes for a
rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with
respect to arclength as $n\to\infty$.
Keywords:Riesz energy, Minimal discrete energy,, Rectifiable curves, Bestpacking on curves Categories:52A40, 31C20 

28. CJM 2001 (vol 53 pp. 1121)
 Athanasiadis, Christos A.; Santos, Francisco

Monotone Paths on Zonotopes and Oriented Matroids
Monotone paths on zonotopes and the natural generalization to maximal
chains in the poset of topes of an oriented matroid or arrangement of
pseudohyperplanes are studied with respect to a kind of local move,
called polygon move or flip. It is proved that any monotone path on a
$d$dimensional zonotope with $n$ generators admits at least $\lceil
2n/(nd+2) \rceil1$ flips for all $n \ge d+2 \ge 4$ and that for any
fixed value of $nd$, this lower bound is sharp for infinitely many
values of $n$. In particular, monotone paths on zonotopes which admit
only three flips are constructed in each dimension $d \ge 3$.
Furthermore, the previously known 2connectivity of the graph of
monotone paths on a polytope is extended to the 2connectivity of the
graph of maximal chains of topes of an oriented matroid. An
application in the context of Coxeter groups of a result known to be
valid for monotone paths on simple zonotopes is included.
Categories:52C35, 52B12, 52C40, 20F55 

29. CJM 2001 (vol 53 pp. 470)
 Bauschke, Heinz H.; Güler, Osman; Lewis, Adrian S.; Sendov, Hristo S.

Hyperbolic Polynomials and Convex Analysis
A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to
a given vector $d$ if the univariate polynomial $t \mapsto p(xtd)$
has all real roots for all vectors $x$. Motivated by partial
differential equations, G{\aa}rding proved in 1951 that the largest
such root is a convex function of $x$, and showed various ways of
constructing new hyperbolic polynomials. We present a powerful new
such construction, and use it to generalize G{\aa}rding's result to
arbitrary symmetric functions of the roots. Many classical and recent
inequalities follow easily. We develop various convexanalytic tools
for such symmetric functions, of interest in interiorpoint methods
for optimization problems over related cones.
Keywords:convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interiorpoint method, semidefinite program, singular value, symmetric function Categories:90C25, 15A45, 52A41 

30. CJM 1999 (vol 51 pp. 1300)
31. CJM 1999 (vol 51 pp. 1258)
 Baake, Michael; Moody, Robert V.

Similarity Submodules and Root Systems in Four Dimensions
Lattices and $\ZZ$modules in Euclidean space possess an infinitude
of subsets that are images of the original set under similarity
transformation. We classify such selfsimilar images according to
their indices for certain 4D examples that are related to 4D root
systems, both crystallographic and noncrystallographic. We
encapsulate their statistics in terms of Dirichlet series
generating functions and derive some of their asymptotic properties.
Categories:11S45, 11H05, 52C07 

32. CJM 1999 (vol 51 pp. 449)
 Bahn, Hyoungsick; Ehrlich, Paul

A BrunnMinkowski Type Theorem on the Minkowski Spacetime
In this article, we derive a BrunnMinkowski type theorem
for sets bearing some relation to the causal structure
on the Minkowski spacetime $\mathbb{L}^{n+1}$. We also
present an isoperimetric inequality in the Minkowski
spacetime $\mathbb{L}^{n+1}$ as a consequence of this
BrunnMinkowski type theorem.
Keywords:Minkowski spacetime, BrunnMinkowski inequality, isoperimetric inequality Categories:53B30, 52A40, 52A38 

33. CJM 1999 (vol 51 pp. 225)
 Betke, U.; Böröczky, K.

Asymptotic Formulae for the Lattice Point Enumerator
Let $M$ be a convex body such that the boundary has positive
curvature. Then by a well developed theory dating back to Landau and
Hlawka for large $\lambda$ the number of lattice points in $\lambda M$
is given by $G(\lambda M) =V(\lambda M) + O(\lambda^{d1\varepsilon
(d)})$ for some positive $\varepsilon(d)$. Here we give for general
convex bodies the weaker estimate
\[
\left G(\lambda M) V(\lambda M) \right 
\le \frac{1}{2} S_{\Z^d}(M) \lambda^{d1}+o(\lambda^{d1})
\]
where $S_{\Z^d}(M)$ denotes the lattice surface area of $M$. The term
$S_{\Z^d}(M)$ is optimal for all convex bodies and $o(\lambda^{d1})$
cannot be improved in general. We prove that the same estimate even
holds if we allow small deformations of $M$.
Further we deal with families $\{P_\lambda\}$ of convex bodies where
the only condition is that the inradius tends to infinity. Here we have
\[
\left G(P_\lambda)V(P_\lambda) \right
\le dV(P_\lambda,K;1)+o \bigl( S(P_\lambda) \bigr)
\]
where the convex body $K$ satisfies some simple condition,
$V(P_\lambda,K;1)$ is some mixed volume and $S(P_\lambda)$ is the
surface area of $P_\lambda$.
Categories:11P21, 52C07 

34. CJM 1998 (vol 50 pp. 426)
35. CJM 1998 (vol 50 pp. 16)
 Böröczky, Károly; Schnell, Uwe

Asymptotic shape of finite packings
Let $K$ be a convex body in $\ed$ and denote by $\cn$
the set of centroids of $n$ nonoverlapping translates
of $K$. For $\varrho>0$, assume that the parallel body
$\cocn+\varrho K$ of $\cocn$ has minimal volume.
The notion of parametric density (see~\cite{Wil93})
provides a bridge between finite and infinite
packings (see~\cite{BHW94} or~\cite{Hen}).
It is known that
there exists a maximal $\varrho_s(K)\geq 1/(32d^2)$ such that
$\cocn$ is a segment for $\varrho<\varrho_s$ (see~\cite{BHW95}).
We prove the existence of a minimal $\varrho_c(K)\leq d+1$ such that
if $\varrho>\varrho_c$ and $n$ is large then
the shape of $\cocn$ can not be too far from the shape of $K$.
For $d=2$, we verify that $\varrho_s=\varrho_c$.
For $d\geq 3$, we present the first example of a convex
body with known $\varrho_s$ and $\varrho_c$; namely, we have
$\varrho_s=\varrho_c=1$ for the
parallelotope.
Categories:52C17, 05B40 

36. CJM 1997 (vol 49 pp. 1162)
 Ku, HsuTung; Ku, MeiChin; Zhang, XinMin

Isoperimetric inequalities on surfaces of constant curvature
In this paper we introduce the concepts of hyperbolic and elliptic
areas and prove uncountably many new geometric isoperimetric
inequalities on the surfaces of constant curvature.
Keywords:Gaussian curvature, GaussBonnet theorem, polygon, pseudopolygon, pseudoperimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalities Categories:51M10, 51M25, 52A40, 53C20 
