The family of rectangles has the cover property, i.e.:
For every $R\geq 1$, $k\geq 1$: every rectangle with aspect ratio $kR$ can be exactly covered by $\lceil k\rceil$ (possibly overlapping) rectangles of aspect ratio at most $R$.
I am trying to generalize this property to arbitrary convex shapes, using the following natural generalization of the aspect ratio concept:
(side length of smallest containing square) / (side length of largest contained square)
This factor can be termed slimness factor, as many papers define an "$R$-fat shape" as a shape for which this factor is at most $R$. So, for example, the slimness factor of a square is 1, the slimness factor of a circle is $\sqrt 2$, etc.
The slimness factor of a rectangle is identical to its aspect ratio, so this is indeed a proper generalization of the aspect ratio concept. However, the cover property is not true when the slimness factor is used instead of the aspect ratio.
For example, taking $R=1$, the only shape with slimness factor 1 is a square, but obviously most convex shapes cannot be exactly covered by a finite number of squares.
So, my question is: Does there exist an $R$ such that every convex shape with slimness factor $kR$ can be exactly covered by $\lceil k\rceil$ convex shapes of slimness factor at most $R$?