This is a weak version of this problem, written down in Lviv Scottish Book.
I start with necessary definitions.
Let $K=-K$ be a centrally symmetric compact convex body in the Euclidean space $\mathbb R^n$, and $\partial K$ be the boundary of $K$ in $\mathbb R^n$.
Definition. A 1-dimensional linear subspace $L\subset \mathbb R^n$ is called an $K$-admissible direction if for every $x\in\partial K$ the intersection $\partial K\cap(x+L)$ contains at most two points. It this case we can define the map $\theta_L:\partial K\to\partial K$ assigning to every point $x\in\partial L$ the unique point $\theta_L(x)$ such that $\partial K\cap(x+L)=\{x,\theta_L(x)\}$.
Remark 1. By the answer to this MO-question, the set of $K$-admissible directions is dense in the space $Gr(1,\mathbb R^n)$ of all 1-dimensional linear subspaces of $\mathbb R^n$.
Problem. Are there $K$-admissible directions $L_1,\dots,L_m$ such that the map $$\theta_{L_m}\circ \dots\circ \theta_{L_1}:\partial K\to\partial K$$ has no fixed points?
Remark 2. For $n=2$ the answer to this problem is affirmative with $m=n=2$.
Example. For the convex body $K=[-1,1]^2$ in $\mathbb R^2$ and the special directions $L_1=\{(x,y)\in\mathbb R^2:x-y=0\}$ and $L_2=\{(x,y)\in\mathbb R^2:x+y=0\}$, the map $\theta_{L_2}\circ\theta_{L_1}:\partial K\to\partial K$ has no fixed points.
