1. CJM 2013 (vol 66 pp. 783)
 Izmestiev, Ivan

Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the HilbertEinstein Functional
The paper is centered around a new proof of the infinitesimal rigidity
of convex polyhedra. The proof is based on studying derivatives of the
discrete HilbertEinstein functional on the space of "warped
polyhedra" with a fixed metric on the boundary.
The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.
In the spherical and in the hyperbolicde Sitter space, there is a perfect duality between the HilbertEinstein functional and the volume, as well as between both kinds of rigidity.
We review some of the related work and discuss directions for future research.
Keywords:convex polyhedron, rigidity, HilbertEinstein functional, Minkowski theorem Categories:52B99, 53C24 

2. CJM 2006 (vol 58 pp. 476)
 Chipalkatti, Jaydeep

Apolar Schemes of Algebraic Forms
This is a note on the classical Waring's problem for algebraic forms.
Fix integers $(n,d,r,s)$, and let $\Lambda$ be a general $r$dimensional
subspace of degree $d$ homogeneous polynomials in $n+1$ variables. Let
$\mathcal{A}$ denote the variety of $s$sided polar polyhedra of $\Lambda$.
We carry out a casebycase study of the structure of $\mathcal{A}$ for several
specific values of $(n,d,r,s)$. In the first batch of examples, $\mathcal{A}$ is
shown to be a rational variety. In the second batch, $\mathcal{A}$ is a
finite set of which we calculate the cardinality.}
Keywords:Waring's problem, apolarity, polar polyhedron Categories:14N05, 14N15 

3. 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 
