For the purposes of this question, define the following properties of convex sets in the plane:

- A set is
**$R$-fat**(for $R\geq 1$) if it contains a disc of side-length $x$ and is contained in a disc of side-length $R\cdot x$, for some positive $x$. - A set is
**$R$-cuttable**if it can be cut (using a straight line) to two non-empty convex $R$-fat shapes.

For example:

- A disc is 1-fat. It is not 1-cuttable since it cannot be cut to two discs. But it is 2-cuttable since it can be cut to two semi-discs, and a semi-disc is 2-fat.
- A $1\times 10$ rectangle is $\sqrt{101}$-fat. It is $\sqrt {26}$-cuttable since it can be cut to two $1\times 5$ rectangles. Hence it is also $\sqrt{101}$-cuttable,

MY QUESTION IS: **what is the smallest value $R_0$ such that every convex $R_0$-fat shape is $R_0$-cuttable?**

The first example shows that $R_0 \geq 2$.

NOTE: I asked a similar question in Math.SE, currently with no reply.

it seems, is into two isosceles triangles, but they are not $R$ fat. Is there a better way to cut it? $\endgroup$