Between $\ell^p$ and $\ell^\infty$ balls?

Question

Take the $\ell^p$ ball in $\mathbb R^d$ of radius $r$ centered at the origin is $$\{x \in \mathbb{R}^d : \sum_{i=1}^d |x_i|^p \le r^p\}.$$

Enclose it by the smallest square possible which is the $\ell^\infty$ ball in $\mathbb R^d$ of radius $r$ centered at origin.

Take a vertex $v$ of the $\ell^\infty$ ball and join the line $L$ between $v$ and origin. $L$ intersects boundary of the $\ell^p$ ball at some point $z$.

1. There are $2^d$ such intersection points. Do the coordinates possess any symmetry?

2. Is there a way to compute intersection coordinates just from symmetry considerations without resorting to analytic geometry?

3. What is the distance from origin to the $z$?

4. Are there any smooth homotopies between $\ell^p$ and $\ell^q$ balls that extends to $p\rightarrow1$ and $q\rightarrow\infty$?