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 $R$**?