Example of homeomorphisms with different supremum metric I hope you can help me. I want to know if there are homeomorphisms $f,g:X\to X$, when $(X,d)$ is compact, such that
$$D(f,g)<D(f^{-1},g^{-1}),$$
where $D(a,b)=\sup_{x\in X}d(a(x),b(x))$.
Any suggestion is welcome :)

Yes. Let $X = [0,4]$. We shall specify piecewise linear homeomorphisms $f, g$ by their graphs in the square $[0,4]^2$. The graph of $f$ consists of the two line segments $F_1$ connecting $(0,0)$ and $(1,2)$ and $F_2$ connecting $(1,2)$ and $(4,4)$. The graph of $g$ consists of the two line segments $G_1$ connecting $(0,0)$ and $(1,3)$ and $G_2$ connecting $(1,3)$ and $(4,4)$. Drawing a picture is helpful. You can easily verify that the distance $D(f,g) = 1$ which is attained at $x=1$. The graphs of $f^{-1},g^{-1}$ are obtained by reflecting the graphs of $f,g$ at the diagonal $y = x$ (draw again a picture). You will see that $\lvert f^{-1}(3) - g^{-1}(3) \rvert > 1$, thus $D(f^{-1},g^{-1}) > 1 = D(f,g)$.
