Please provide some references on Voronoi and Delaunay decompositions which is mathematically written. I mean I can find several texts or links on this written for computer science students without any mathematical theorems etc. It would be great if there is some text or article with connection to toric varieties.
There is a lot of background material on Voronoi and Delaunay in the paper of Oda and Seshadri. But it is briefly written. I wanted some elaborate article on this basic material.
I'd recommend the following two books. In the first one there are some connections to toric varieties. (Chapter 9.3: Lattice polytopes and unimodular triangulations; also they deal with regular aka coherent triangulations.)
De Loera, Jesús A.; Rambau, Jörg; Santos, Francisco, Triangulations. Structures for algorithms and applications, Algorithms and Computation in Mathematics 25. Berlin: Springer (ISBN 978-3-642-12970-4/hbk; 978-3-642-12971-1/ebook). xiii, 535 p. (2010). ZBL1207.52002.
Aurenhammer, Franz; Klein, Rolf; Lee, Der-Tsai, Voronoi diagrams and Delaunay triangulations, Hackensack, NJ: World Scientific (ISBN 978-981-4447-63-8/hbk). viii, 337 p. (2013). ZBL1295.52001.
Maybe this PhD thesis could help:
Anton, Francois. Voronoi diagrams of semi-algebraic sets. Diss. University of British Columbia, 2003. PDF download.
"The theoretical purpose of this thesis is to elucidate the basic algebraic and geometric properties of the offset to an algebraic curve and to reduce the semialgebraic computation of the Delaunay graph to eigenvalues computations. The practical objective of this thesis is the certified computation of the Delaunay graph for low degree semi-algebraic sets embedded in the Euclidean plane."
Fig.3.1.4
He uses toric varieties (p.71ff).
This question is much too broad. However, a good introduction (to various generalizations, as well) is in Edelsbrunner's little book.
Edelsbrunner, Herbert, Geometry and topology for mesh generation., Cambridge Monographs on Applied and Computational Mathematics 7. Cambridge: Cambridge University Press (ISBN 0-521-68207-X/pbk). xii, 177 p. (2006). ZBL1088.65014.
For connections to polytopes, you can see Paco Santos' 2006 ICM talk, which is extremely well-written (Santos is occasionally active on MO, so might add something else).
