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?
Best Answer
Any
simply connectedpolygon must be star-shaped to be shrinkable. I have made minor edits below to treat the more general case.Let $D$ be a polygon with convex hull $H$. Assume we are given a non-trivial shrinking of $D$; view this as a map from $H$ to itself. This map must have a fixed point $x$, either by algebraic topology or an iterative construction.
This means it suffices to consider only dilations centered at a point $x$ in $H$, rather than dilations followed by translations.
For any $x$, if there is a point $y$ in $D$ so that the segment from $x$ to $y$ is not contained in $D$, then a $(1-\epsilon)$-dilation of $H$ centered at $x$ will not carry $D$ into $D$ for any positive $\epsilon$ smaller than some $\epsilon(x)>0$. If $D$ is not star-shaped, take the minimum $\delta$ of $\epsilon(x)$ over $x\in H$, and then no $(1-\delta)$-dilation of $H$ centered at a point in $H$ carries $D$ into $D$.