10.09.03. We introduced the notions of an affine monoid M and its group of differeneces gp(M). We proved the Gordan Lemma (that the monoid of lattice points in a rational finite cone is finitely gerenereted). This result remains true for arbitrary submonoid of Z^d which spans a finite cone in R^d. The smallest generating set of these monoids exists and it coincides with the set of indecomposable elements of the monoid. For monoids of the lattice points in a simplicial cone the Hilbert basis is a subset of the parallelotope spanned by the extremeal generators. Section 2.A: Lemma 2.7, Corollaries 2.8, 2.9, Subsection: Positive Monoids
10.21.03. We introduced the notion of  the bottom of a positive cone C and observed that lattice points on the bottom always belong to Hilb(C). In dimension 2 they even exhaust all of Hilb(C). This gives rise to a Unimodular Hilbert Triangulation of a positive two-dimensional cone. In general, we have introduced the concepts of  an X-triangulation (full X-triangulation) of conv(X) for a finite subset X of R^d as well as a Y-triangulation (full Y-triangulation) of a cone cone(L_1,...,L_n) where Y={L_1,...,L_n} is a finite family of rays in R^d whose conical set cone(L_1,...,L_n) is a positive cone. (This terminology was only used in class, not in Polytopes, Rings, and K-Theory). Using convex functions and projections we proved that full X-triangulations always exist and derived the similar claim for cones. The triangulations obtained this way are colled regular. Not all traingualations are regular. Sections 1.F (containing much more information, ignore general polyhedral complexes and think of these complexes as triangulations of a single polytope) and the beginning of Subsection Pointed Rational Cones in Section 2.C (also containing mach more information, you just assume L=Z^d).
It becomes increasingly important that you take notes in the class because what I talk about is a fraction of what is presented in Polytopes, Rings, and K-Theory.
10.23.03. We introduced the notion of the Caratheodory rank CR(C) of a cone C (rational, positive). It is the maximum of the numbers of different elements of Hilb(C) one needs to represent arbitrary lattice point in C as a non-negative integral linear combination. For a unimodular cone C we have CR(C) = dim C and in dimension 2 we have CR(C)=2. In general, Sebo's theorem gives the estimates dim C < or = CR(C) < or = 2 dim C - 2. The proof is through reduction (via full triangulations determind by the bottom of C) to the special case when the cone is simplicial and, moreover, the simplex spanned by the origin and the extremal generators is empty. Second part of the Subsection Pointed Rational Cones in Section 2.C.
10.28.03. We have the table of what is known up to date:

dim(C)UHT  |  UHCICP
    1      |   yes   |  yes     |  yes
    2      |   yes   |  yes     |  yes
    3      |   yes   |  yes     |  yes
    4      |   no    |     ?     |     ?
    5      |    ?     |     ?     |     ?
    6      |   no    |    no    |    no

Also, in every dimension (UHT) ==> (UHC) ==> (ICP). Starting from dimension four (UHC
) does not imply (UHT). It is not known whether there are cones which have (ICP) but violate (UHC).

Today we proved that arbitrary rational positive cone admits a unimodular triangulation. We also sketched the proof of tha fact that 3-dimensional rational positive cones have (UHT). Section 2.D, Theorem 2.58 and Theorem 2.61.
10.30.03. All polyotpes in this discussion are assumed to be lattice. A polytope P is called "normal" if the lattice point in its shifted copy (P,1) in hight 1 constitue the Hilbert basis of the homogenization cone C(P)={ a x :  a> or = 0 and  x=(y,1) for some y\in P }. Uimodular simplices are normal, as well as  parallelotopes.  If a polytope is covered by  normal polytopes then it is normal. In particular, polytopes that are covered by unimodular simplices are normal. All polygons (dim = 2). But starting from dimension 5 there are normal polytopes that are not covered by unimodular simlices. An example was found by an extensive computer search at turned out to be very aesthetic polytope: it is empty, highly simmetric, etc. In dimension 3 and 4 it is not known whether there are normal polytopes that are not unimodularly covered. Subsection Tight Cones in Section 2.D.
11.04.03. We proved that for any lattice polytope P there is a  natural number c_0 such that the dilated polytopes cP are covered by unimodular simplices whenever c > or = c_0. The proof uses two facts: (1) any  positive rational cone has a unimodular triangulation and (2) the standard unimodular cube has a unimodular cover (even a triangulation). The latter claim is proved by constructing explicit triangulations: we first order the vertices of the standard unit cube so that v < w if there is a standard basic vector e_i such that v + e_ i = w and then form the simplices in the cube by taking the convex hulls of the chains in this partially ordered set. For a detailed proof go to
this paper and click on PDF file, it is Theorem 3.4.1.

The triangulation we obtain for the unit cube has the nice property that it restrict to the same triangulations on opposite facets up to a paralle translate. In particular, the shifted copies of these triangulations in any rectangular parallelotope patch up to a global triangulation of the rectangular parallelotope. Another consequence is that multiples (i. e. dilated copies) of a unimodular simplex have unimodular triangulations.
11.06.03. We introduced the notion of a polyhedral complex. This is a global geometric object built out of usual polytopes much like a topological manifold is built out of copies of R^n. An isomorphism of two polyhedral complexes is defined in a natural way. In the special case when a polyhedral complex consists of polytopes (and their faces) in some ambient Euclidean space the complex is called embedded, and a complex that can isomorphically made embedded is called embeddable. Not all polyhedral complex are embeddable, but simplicial complexes (i. e. complexes whose faces are simplices) are always embeddable. Section 1.D.
11.13.03. We introduced the notion of a polyhedral subdivision and that of a regular polyhedral subdivision. When the faces of a subdivision are simplices the subdivision is called a triangulation. Regular subdivisions are defined in terms of support functions. We mainly talked on the special case of subdivisions of a singhle polytope (meaning the complex of all faces of a single polytope). The set of support finctions for a subdivision \Pi forms in a natural way the relative interion of a finite pointed cone in the space R^n where n is the number of vertices of \Pi. When \Pi is a triangulation this cone has dimension n.

A dissection of a polytope or, more generally, manyfold dissections by hyperplanes produce a regular subdivision.
If \Pi is a regular subdivision of a polytope P and for each face F \in \Pi we are given a regular subdivision \Pi_F then the subdivisions \Pi_F patch up to a regular subdivision of P if there is a choice of support functions h_F \in SF(\Pi_F), F runs through the faces of \Pi, such that h_F and h_G restrict to the same function on the intersection of F and G for arbitrary faces F and G of \Pi.

We introduced the notion of a fan: it is an embedded conical complex. A complete fan is a fan whose support space is all of the ambient space
R^d (equivalently, it is a conical subdivision of R^d). A complete fan is called projective if it is a regular subdivision of the ambient space. A complete fan is projective if and only if either of the following two cinditions is satisfied: (a) it is a normal fan N(P) of some d-dimensional polytope P, and (b) it is a fan of the cones over faces of some polytope whose interior contains the origin 0 \in R^d. Section 1.F.
11.18.03. We considered Weyl chambers in dimension d, indexed by the elements of the permutation group S(d). They triangulate the unit d-cube and their parallel translates by integral vectors tile the entire space R^d. This tiling can be also obtained by dissecting R^d by certain family of hypeprlanes. The vertex sets of Weyl chambers are naturally ordered and the orderings are compatible - the vertex set of the inetersection of parallel translates by integral vectors of two Weyl chambers carry the same order in the both copies of the chambers. Section 3.A, up to Lemma 3.3 (including).
11.20.03. We introduced the concept of a mixed triangulation of a simplicial complex. Rest of Section 3.A.
11.25.03. We introduced the notion of a lattice polyhedral complex. Second half of Section 1.G.
We completed a brief description of the main ideas in the proof of Knudson-Mumfod theorem on multiples of lattice polytopes. In the final week we will discuss further combinatorial/algebraic aspects of lattice polytopes, not treated in Part 1 of Polytopes, Rings, and K-Theory.