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?

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):

Formally, a compact $\ P\subseteq \mathbb R^n\ $ is called shrinkable iff:

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

What is the largest group of shrinkable polygons?

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

Proof: By definition of a star-shaped polygon, there exists a point $A\in P$ such that for every $B\in P$, the segment $AB$ is entirely contained in $P$. Now, for all $\mu\in [0;1)$, let $\ q := (1-\mu)\cdot A$. This effectively translates the dilated $P'$ such that $A'$ coincides with $A$. Now every point $B'\in P'$ is on a segment between $A$ and $B$, and hence contained in $P$.

My questions are:

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

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

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?

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

Proof: By definition of a star-shaped polygon, there is a point $A\in P$ such that for all other points $B\in P$, the segment $AB$ is entirely contained in $P$. Translate the dilated polygon $P'$ such that $A'$ coincides with $A$ (its counterpart in $P$). Now every point $B' \in P'$ is on the segment $AB$, hence it is in $P$:

My questions are:

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

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

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?