I have an open bounded convex set in $\mathbb{R}^n,$ $A.$ Fix $x\in A,$ and consider the function $\phi:S^{n-1}\to\mathbb{R}$ such that, for all $e\in S^{n-1},$ $\phi(e)=\sup\{t\geq 0: x+te\in A\}.$ I have proved that this function is continuous, and that $\partial A=\{x+\phi(e)e: e\in S^{n-1}\}.$ I want to prove that $\phi$ is Lipschitz continuous, and I want to estimate the Lipschitz constant using the diameter of $A$ and the distance of $x$ from $\partial A.$
I have $d(x,\partial A)=\inf_{y\in\partial A}{d(x,y)}=\inf_{e\in S^{n-1}}\phi(e),$ and $diam(A)=\sup_{x,y\in A}{d(x,y}).$ I think the Lipschitz constant depends on $\frac{d(x,\partial A)}{diam(A)}$ (maybe this is the Lipschitz constant?), but I’m stuck in proving this fact.