  Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between
boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family
of probability measures $\{ \mu_x \}_{x \in X}$ on $\partial X$, we describe a
continuous family of extensions $\{\hat{f}_p : X \to Y \}_{1 \leq p \leq
\infty}$ of $f$, called the hyperbolic $p$-barycenter maps of $f$. If all the
measures $\mu_x$ have full support then for $p = \infty$ the map
$\hat{f}_{\infty}$ coincides with the circumcenter map $\hat{f}$ defined
previously in \cite{biswas5}. We use this to show that if $X, Y$ are complete,
simply connected manifolds with sectional curvatures $K$ satisfying $-b^2 \leq
K \leq -1$, then the circumcenter maps of $f$ and $f^{-1}$ are
$\sqrt{b}$-bi-Lipschitz homeomorphisms which are inverses of each other. It
follows that closed negatively curved manifolds with the same marked length
spectrum are bi-Lipschitz homeomorphic.
