Let's call a polygon $P$ shrinkable if any down-scaled (dilated) version of $P$ can be translated into $P$. For example, the following triangle is shrinkable (the original polygon is green, the dilated polygon is blue):

But the following U-shape is not shrinkable (the blue polygon cannot be translated into the green one):

What is the largest group of shrinkable polygons?

More generally, a compact $\ P\subseteq \mathbb R^n\ $ is called shrinkable $\ \Leftarrow:\Rightarrow\ $

$$\forall_{\mu\in [0;1)}\ \exists_{q\in \mathbb R^n}\quad \mu\!\cdot\! P\, +\, q\ \subseteq\ P$$

Currently I have the following shrinkable property sufficient condition: if $P$ is star-shaped then it is shrinkable.

Proof: By definition (just for the star vertex $\ A\in P,\ $ let $\ q := (1-\mu)\cdot A$) :

My questions are:

A. Is the condition of being star-shaped also necessary for shrinkability?

B. Alternatively, what other condition on $P$ is necessary?