Consider a set $G$ of the form $$G=\{ru:u\in\mathbb S^{d-1}, 0\leq r\leq \phi(u)\},$$ Where $\mathbb S^{d-1}$ is the unit sphere in $\mathbb R^d$ and $\phi:\mathbb S^{d-1}\to [0,\infty)$ is a given function. Is it true that if $G$ is convex, then $\phi$ needs to be Lipschitz ?
Thank you !