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A triangulation of a polytope into simplices is refined recursively. In everyrefinement round, some simplices which have been marked by an externalalgorithm are bisected and some others around also must be bisected to retainregularity of the triangulation. The ratio of the total number of markedsimplices and the total number of bisected simplices is bounded from above.Binev, Dahmen and DeVore proved under a certain initial condition a bound thatdepends only on the initial triangulation. This thesis proposes a new way toobtain a better bound in any dimension. Furthermore, the result is proven for aweaker initial condition, invented by Alk\"amper, Gaspoz and Kl\"ofkorn, whoalso found an algorithm to realise this condition for any regular initialtriangulation. Supposably, it is the first proof for a Binev-Dahmen-DeVoretheorem in any dimension with always practically realiseable initial conditionswithout an initial refinement. Additionally, the initialisation refinementproposed by Kossaczk\'y and Stevenson is generalised, and the number ofrecursive bisections of one single simplex in one refinement round is boundedfrom above by twice the dimension, sharpening a result of Gallistl, Schedensackand Stevenson.

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