From the text: This thesis deals with unimodular covers and triangulations of lattice polytopes. It centers around the following problem definition: Provide a good bound \(c_d\) such that for all \(d\)-dimensional lattice polytopes and \(c \geq c_d\) all multiples \(cP\) admit a unimodular cover.
In [Toroidal embeddings. I. York: Springer-Verlag (1973; Zbl 0271.14017)], F. Knudsen and D. Mumford showed that for every lattice polytope \(P\) there exists a number \(c_P\) such that \(c_P P\) admits a unimodular triangulation. One might ask if this result can be generalized. Is there a number \(c_d\) only depending on the dimension of the lattice polytope \(P\) such that the multiples \(c P\) admit a unimodular triangulation for all \(d\)-dimensional lattice polytopes \(P\) and \(c \geq c_d\)?
In the second chapter we will improve the bound by modifying a crucial step (with respect to the numerical result) in the proof of the above statement. This crucial step is based on a simple procedure for covering simplicial cones by unimodular cones. We will provide a new procedure for covering simplicial cones, which is better than the original one in a sense that the new procedure gives us a cover with unimodular cones whose generators are relatively short.
In the third chapter we will provide a similar procedure for the unimodular triangulation of simplicial cones which has no consequences on the bounds \(c_d^{\mathrm{cone}}\) and \(c_d^{\mathrm{pol}}\), but might be of interest itself.
In the fourth chapter we turn away from multiples of polytopes and focus on stellar subdivisions of lattice polytopes. This turn is motivated by the fact that all procedures and results in the previous chapters are, roughly speaking, due to the successive application of stellar subdivision. Therefore, we speculate about the importance of this tool for the triangulation of lattice polytopes.