Packing rectangles into a rectangle
In the sequel, it is assumed that the sides of the geometrical objects considered are parallel. \textit{J. Januszewski} [Geom. Dedicata 81, 13--18 (2000; Zbl 0959.52013)] proved that any sequence of rectangles of side lengths at most 1 and whose total area is at most \(\frac{1}{2}\) can be packed into a unit square. This result is here extended by showing that any sequence of rectangles of side lengths at most \(a\) and whose total area is at most \(\frac{ab}{2}\) can be packed into an \(a\times b\) rectangle, where \(a \leq b\). The bound in the result cannot be tightened: for any \(\varepsilon >0\), three squares of side \(1+\varepsilon\) cannot be packed into a \(2\times 3\) rectangle.
scientific article; zbMATH DE number 2119035