Two-dimensional partitions (Q1332593)

Given a positive function \(\delta\) on a rectangle \(R\) and some \(\lambda\in ]0, 1]\), one says that \(R\) can be properly partitioned if one can find a finite collection of nonoverlapping subrectangles \(R_ j\), \(1\leq j\leq m\), with union \(R\), with regularity (ratio of the smaller side on the large one) greater or equal to \(\lambda\), and one vertex \(x_ j\) of \(R_ j\) such that each side of \(R_ j\) is smaller or equal to \(\delta(x_ j)\). In other words, there exists a \(\lambda\)-regular \(\delta\)-finite tagged-partition with the tags at one edge of the associated rectangle. Buczolich had proved that if \(\lambda= 1/1000\), and \(\delta\) is arbitrary, then every rectangle can be properly partitioned. The result is extended in this paper to every \(\lambda\in ]0, 1/\sqrt 2[\), and even to \(\lambda= 1/\sqrt 2\) when \(\delta\) is upper semicontinuous.