Expand all Collapse all | Results 26 - 35 of 35 |
26. CJM 2004 (vol 56 pp. 472)
Infinite-Dimensional 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
finite-dimensional affine sections is a (standard) polytope.
We study the relationships between eight known definitions
of infinite-dimensional
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 2001 (vol 53 pp. 1121)
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
pseudo-hyperplanes 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 $\lceil
2n/(n-d+2) \rceil-1$ flips for all $n \ge d+2 \ge 4$ and that for any
fixed value of $n-d$, 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 2-connectivity of the graph of
monotone paths on a polytope is extended to the 2-connectivity 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 |
28. CJM 2001 (vol 53 pp. 470)
Hyperbolic Polynomials and Convex Analysis A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to
a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$
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 convex-analytic tools
for such symmetric functions, of interest in interior-point methods
for optimization problems over related cones.
Keywords:convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interior-point method, semidefinite program, singular value, symmetric function Categories:90C25, 15A45, 52A41 |
29. CJM 1999 (vol 51 pp. 1258)
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 self-similar images according to
their indices for certain 4D examples that are related to 4D root
systems, both crystallographic and non-crystallographic. We
encapsulate their statistics in terms of Dirichlet series
generating functions and derive some of their asymptotic properties.
Categories:11S45, 11H05, 52C07 |
30. CJM 1999 (vol 51 pp. 1300)
On the Existence of Similar Sublattices Partial answers are given to two questions. When does a lattice
$\Lambda$ contain a sublattice $\Lambda'$ of index $N$ that is
geometrically similar to $\Lambda$? When is the sublattice
``clean'', in the sense that the boundaries of the Voronoi cells
for $\Lambda'$ do not intersect $\Lambda$?
Category:52C07 |
31. CJM 1999 (vol 51 pp. 449)
A Brunn-Minkowski Type Theorem on the Minkowski Spacetime In this article, we derive a Brunn-Minkowski 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
Brunn-Minkowski type theorem.
Keywords:Minkowski spacetime, Brunn-Minkowski inequality, isoperimetric inequality Categories:53B30, 52A40, 52A38 |
32. CJM 1999 (vol 51 pp. 225)
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 $G(\lambda M) =V(\lambda M) + O(\lambda^{d-1-\varepsilon
(d)})$ for some positive $\varepsilon(d)$. Here we give for general
convex bodies the weaker estimate
\[
\left| G(\lambda M) -V(\lambda M) \right |
\le \frac{1}{2} S_{\Z^d}(M) \lambda^{d-1}+o(\lambda^{d-1})
\]
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 $o(\lambda^{d-1})$
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 |
33. CJM 1998 (vol 50 pp. 426)
34. CJM 1998 (vol 50 pp. 16)
Asymptotic shape of finite packings Let $K$ be a convex body in $\ed$ and denote by $\cn$
the set of centroids of $n$ non-overlapping 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 |
35. CJM 1997 (vol 49 pp. 1162)
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, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalities Categories:51M10, 51M25, 52A40, 53C20 |