Let $\gamma:I\subset\mathbb{R}\rightarrow\mathbb{R}^2, s\mapsto \gamma(s)$, denote the arclength parametrization of a smooth, convex curve $\Gamma:=\gamma(I)\subset\mathbb{R}^2$. Equip $\Gamma$ with arclength measure $\sigma$.

Here is my question: what extra conditions on $\Gamma$ will ensure that the triple convolution $\sigma\ast\sigma\ast\sigma$ is a *smooth* function of $x\in\mathbb{R}^2$ inside its support (i.e. the set $\Gamma+\Gamma+\Gamma\subset\mathbb{R}^2$), up to the boundary? I am also interested in partial regularity results, and counterexamples.

Here is a possible approach: for any $x\in\mathbb{R}^2$ and $\epsilon>0$, $$\int_{|y-x|<\epsilon} \sigma\ast\sigma\ast\sigma(y)dy=|\{(s,s',s'')\in I^3: |\gamma(s)+\gamma(s')+\gamma(s'')-x|<\epsilon\}|.$$

Understanding how these sets look like will probably help. I am familiar with results in this spirit for *double* convolutions (e.g. Fefferman 1970), but not for triple ones.

References would be much appreciated.

Thank you.