Context. I'm working on a problem, and it seems Steiner symmetrization might just be the golden trick. But first, I must make sure the process will converge to an $\ell_p$ ball...
Fix $p \in [1,\infty]$ and let $\|\cdot\|_p$ be the $\ell_p$-norm on $\mathbb R^n$. Let $B_{n,p}(r) := \{x \in \mathbb R^n \mid \|x\|_p \le r\}$ be the $\ell_p$-ball in $\mathbb R^n$, with radius $r \ge 0$. Let $\mathbb K_n$ be the set of all compact convex subsets of $\mathbb R^n$ which contain the origin in their interior, and let $K \in \mathbb K_n$, nonempty. For any nonzero vector $u$ in $\mathbb R^n$, one can view $K$ as a family of finite line segments parallel to $u$. slide each line segment so that it is symmetric about the hyperplane $u^\perp := \{x \in \mathbb R^n \mid x^Tu = 0\}$. Denote the so obtained set as $S_uK$. Note that $S_uK \in \mathbb K_n$.
Question. For which sets $K \in \mathbb K_n$ is it possible to find $r \ge 0$ and a sequence $u_1,u_2,\ldots,u_N,\ldots$ of nonzero vectors in $\mathbb R^n$ such that one has (in Hausdorff topology, for example) $$ \lim_{N \rightarrow \infty}S_{u_N}S_{u_{N-1}}\ldots S_{u_2}S_{u_1}K = B_{n,p}(r)\; ? $$
Notes
- The euclidean case $p=2$ is solved by Theorem 9.1 of "Convex and Discrete Geometry" (Peter M. Gruber) .
- I'm particularly interested in the cases $p=1$ and $p=\infty$.