Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping $$ f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X, $$ with $f(0)=0$ is a radial and maps the unit ball of $(X,\|\cdot\|_a)$ onto the unit ball of $(X,\|\cdot\|_b)$.
We can assume that the two norms are equivalent.
What is the smallest Lipschitz constant of $f$? That is, what is the smallest constant $L \geq 0$ in the following inequality: $$ \| f(x) - f(y) \|_b \leq L \| x - y \|_a $$ It is possible to get an estimate in terms of the equivalence constants of the norms (and along rays from the origin, it is just an equivalence constant) but I am explicitly looking for the most tight estimates known.
