Assume that $M\subset R^n$, $n\ge 3$, is a boundary of an open bounded set $D$ containing $0$, which is starlike w.r.t. 0, meaning that each ray $[0,x]$ from $x\in M$ to $0$ meets $M$ only once. Is $M$ smooth almost everywhere?
The unit circle is a boundary of an open bounded set $D$ containing $\mathbf{0}$,
is a boundary of an open bounded set $D$ containing $\mathbf{0}$, which


