In joint work with Joerg Rambau and Francisco Santos, I wrote a book on
Triangulations, Triangulations: Structures for Algorithms and
Applications which is volume 25 of the series Algorithms and
Computation in Mathematics of Springer Verlag. It appeared in 2010
(539p. 496 illus., 281 illus. in color!). We think it contains most of
the key results related to the theory of secondary polytopes, regular
triangulations, flips, and related subjects. Triangulations of
polyhedra appear in combinatorics, commutative algebra, optimization,
and other subjects.
Jointly with Raymond Hemmecke and Matthias Koeppe, I wrote a book on
the emerging theory of algebraic techniques in optimization. The title is
Algebraic and Geometric Ideas in the Theory of Discrete
Optimization and was published as volume 14 of the SIAM and
Mathematical Optimization Society joint Series on Optimization. We
hope to promote these algebraic ideas to more optimization
workers. The book shows several applications of algebra and geometry
to problems with non-linear constraints and hopefully demonstrates the
need to use algebraic ideas in the transition from linear discrete
optimization to nonlinear discrete optimization.
Both books are available as E-books, and often are freely available like
that to all your students if your library subscribes to Springer and SIAM journals.
Here is a link to our popular software LattE for counting
integer lattice points in convex polytopes and computing Ehrhart
polynomials. In its latest version, you can also compute integrals of
polynomials over a rational polytope (exact), in particular it can do volumes.
The program universalbuilder,
written together with
Samuel Peterson, is a C++ program that generates CPLEX
readable integer programs to compute minimal and maximal
triangulations of convex polytopes of arbitrary dimension.
It can also compute chirotopes of point configurations.
The program PUNTOS allows you to compute
regular triangulations of point sets. It also has useful subroutines
for checking regularity and computing the GKZ coordinates of the secondary
polytope. BUT you should REALLY use TOPCOM instead!