convexity - Determination of a symmetric convex region by parallel sections

Yes.

Assume we have two distinct functions $f$ and $g$ such that $f^{-1}-fequiv g^{-1}-g$.
Take a sequence $x_n=f(x_{n-1})$.
Clearly $f(x_n)-g(x_n)=0$ or $(-1)^n[f(x_n)-g(x_n)]$ has the same sign for all $n$.

Sinse $int_0^1f=int_0^1g$, there are two sequences $x_n$ and $y_n$ as above such that $f(x_n)=g(x_n)$, $f(y_n)=g(y_n)$ and say $(-1)^n[f(x)-g(x)]>0$ for any $xin(x_n,y_n)$.
Note that $x_n,y_nto 0$ and $int_{x_n}^{y_n}|f-g|=const>0$.
It follows that $limsup_{xto0} |f(x)-g(x)|toinfty$, a contradiction.