Let $C \subset \Bbb R^d$ be a Lebesgue-measurable subset such that $C+C = \{x+y \mid x,y\in C\}$ is measurable.
What can we say about the measure (the volume) of $C+C$ ? I know that if $C$ is convex, then $C+C=2C$, so $m(C+C)=2^d m(C)$. Is this still true if $C$ is star-shaped, i.e. $[x,y] \subset C$ for every $x,\in C$? This is the most interesting case to me.
(Some other questions are: What are some other sufficient conditions to have $m(C+C)=2^d m(C)$ ? I know that $2C \subset C+C$ so $2^d m(C) \leq m(C+C)$ always holds).
Take $C = ([-1,1]\times \{0\} ) \cup ( \{0\}\times [-1,1])$. This is star-shaped with respect to the origin and has measure zero, but $C + C \supset [-1,1]^2$.