1. CJM 2016 (vol 69 pp. 1143)
 SikiriÄ‡, Mathieu Dutour

The seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices
For a lattice $L$ of $\mathbb{RR}^n$, a sphere $S(c,r)$ of center $c$
and radius $r$
is called empty if for any $v\in L$ we have $\Vert v 
c\Vert \geq r$.
Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay
polytope}
$P=\operatorname{conv}(S(c,r)\cap L)$.
A Delaunay polytope is called {\em perfect} if any affine transformation
$\phi$ such that $\phi(P)$ is a Delaunay polytope is necessarily
an isometry
of the space composed with an homothety.
Perfect Delaunay polytopes are remarkable structure that exist
only
if $n=1$ or $n\geq 6$ and they have shown up recently in covering
maxima studies.
Here we give a general algorithm for their enumeration that relies
on
the Erdahl cone.
We apply this algorithm in dimension $7$ which allow us to find
that there are only two perfect Delaunay polytopes: $3_{21}$
which
is a Delaunay polytope in the root lattice $\mathsf{E}_7$ and
the
Erdahl Rybnikov polytope.
We then use this classification in order to get the list of all
types
Delaunay simplices in dimension $7$ and found $11$ types.
Keywords:Delaunay polytope, enumeration, polyhedral methods Categories:11H06, 11H31 

2. CJM 2011 (vol 63 pp. 1254)
 D'Azevedo, Antonio Breda; Jones, Gareth A.; Schulte, Egon

Constructions of Chiral Polytopes of Small Rank
An abstract polytope of rank $n$ is said to be chiral if its
automorphism group has precisely two orbits on the flags, such that
adjacent flags belong to distinct orbits. This paper describes
a general method for deriving new finite chiral polytopes from old
finite chiral polytopes of the same rank. In particular, the technique
is used to construct many new examples in ranks $3$, $4$, and $5$.
Keywords:abstract regular polytope, chiral polytope, chiral maps Categories:51M20, 52B15, 05C25 

3. CJM 2010 (vol 62 pp. 1293)
 Kasprzyk, Alexander M.

Canonical Toric Fano Threefolds
An inductive approach to classifying all toric Fano varieties is
given. As an application of this technique, we present a
classification of the toric Fano threefolds with at worst canonical
singularities. Up to isomorphism, there are $674,\!688$ such
varieties.
Keywords:toric, Fano, threefold, canonical singularities, convex polytopes Categories:14J30, 14J30, 14M25, 52B20 

4. CJM 2009 (vol 61 pp. 1300)
 Hubard, Isabel; Orbani\'c, Alen; Weiss, Asia Ivi\'c

Monodromy Groups and SelfInvariance
For every polytope $\mathcal{P}$ there is the universal regular
polytope of the same rank as $\mathcal{P}$ corresponding to the
Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given
automorphism $d$ of $\mathcal{C}$, using monodromy groups, we
construct a combinatorial structure $\mathcal{P}^d$. When
$\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that
$\mathcal{P}$ is selfinvariant with respect to $d$, or
$d$invariant. We develop algebraic tools for investigating these
operations on polytopes, and in particular give a criterion on the
existence of a $d$\nobreakdashauto\morphism of a given order. As an application,
we analyze properties of selfdual edgetransitive polyhedra and
polyhedra with two flagorbits. We investigate properties of medials
of such polyhedra. Furthermore, we give an example of a selfdual
equivelar polyhedron which contains no polarity (duality of order
2). We also extend the concept of Petrie dual to higher dimensions,
and we show how it can be dealt with using selfinvariance.
Keywords:maps, abstract polytopes, selfduality, monodromy groups, medials of polyhedra Categories:51M20, 05C25, 05C10, 05C30, 52B70 

5. CJM 2005 (vol 57 pp. 844)
 Williams, Gordon

Petrie Schemes
Petrie polygons, especially as they arise in the study of regular
polytopes and Coxeter groups, have been studied by geometers and group
theorists since the early part of the twentieth century. An open
question is the determination of which polyhedra possess Petrie
polygons that are simple closed curves. The current work explores
combinatorial structures in abstract polytopes, called Petrie schemes,
that generalize the notion of a Petrie polygon. It is established
that all of the regular convex polytopes and honeycombs in Euclidean
spaces, as well as all of the Gr\"unbaumDress polyhedra, possess
Petrie schemes that are not selfintersecting and thus have Petrie
polygons that are simple closed curves. Partial results are obtained
for several other classes of less symmetric polytopes.
Keywords:Petrie polygon, polyhedron, polytope, abstract polytope, incidence complex, regular polytope, Coxeter group Categories:52B15, 52B05 

6. CJM 1999 (vol 51 pp. 1230)
7. CJM 1999 (vol 51 pp. 1240)