  A set of piecewise linear functions, called polylines, $P_1,\ldots,P_L$ each
with at most $n$ vertices can be simplified into a polyline $M$ with $k$
vertices, such that the Fr\'echet distances $\epsilon_1,\ldots,\epsilon_L$ to
each of these polylines are minimized under the $L_p$ distance. We call $M$ for
$L_p$ with $p\geq 1$ a $p$-mean curve ($p$-MC).
  We discuss $p\geq 1$, for which $L_p$ distance satisfies the triangle
inequality and $p$-mean has not been discussed before for most values $p$.
Computing the $p$-mean polyline is NP-hard for $L=\Omega(1)$ and some values of
$p$, so we discuss approximation algorithms.
  We give a $O(n^2\log k)$ time exact algorithm for $L=2$ and $p\geq 1$. Also,
we reduce the Fr\'echet distance to the discrete Fr\'echet distance which adds
a factor $2$ to both $k$ and $\epsilon$. Then we use our exact algorithm to
find a $3$-approximation for $L>2$ in $\operatorname{poly}(n,L)$ time. Our
method is based on a generalization of the free-space diagram (FSD) for
Fr\'echet distance and composable core-sets for approximate summaries.
