Let $P$ be a convex polygon, and $x \in P$ a point inside $P$. Reflect $x$ in each edge of $P$, and connect the reflections in the same order as the edges, producing a new polygon $P^*$. For example:
The blue polygon is $P$. The red polygon is $P^*$.
Note that $P^*$ is not necessarily convex; nor is it always true that $P^* \supset P$.
Does this reflected object have a name in the literature? Perhaps for (smooth) convex bodies, and perhaps in arbitrary dimensions?
It could as well be defined for nonconvex $P$. I would name it the reflect of $P$ with respect to $x$. This object has come up in my work, and it would be useful to know if there is any literature on its properties.