The Constant in the Theorem of Binev-Dahmen-DeVore-Stevenson and a Generalisation of it

A triangulation of a polytope into simplices is refined recursively. In every refinement round, some simplices which have been marked by an external algorithm are bisected and some others around also must be bisected to retain regularity of the triangulation. The ratio of the total number of marked simplices and the total number of bisected simplices is bounded from above. Binev, Dahmen and DeVore proved under a certain initial condition a bound that depends only on the initial triangulation. This thesis proposes a new way to obtain a better bound in any dimension. Furthermore, the result is proven for a weaker initial condition, invented by Alk\"amper, Gaspoz and Kl\"ofkorn, who also found an algorithm to realise this condition for any regular initial triangulation. Supposably, it is the first proof for a Binev-Dahmen-DeVore theorem in any dimension with always practically realiseable initial conditions without an initial refinement. Additionally, the initialisation refinement proposed by Kossaczk\'y and Stevenson is generalised, and the number of recursive bisections of one single simplex in one refinement round is bounded from above by twice the dimension, sharpening a result of Gallistl, Schedensack and Stevenson.