  For $1 \leq p < \infty$, Ludwig, Haberl and Parapatits classified $L_p$
Minkowski valuations intertwining the special linear group with additional
conditions such as homogeneity and continuity. In this paper,a complete
classification of $L_p$ Minkowski valuations intertwining the special linear
group on polytopes without any additional conditions is established for $p \geq
1$ including $p = \infty$. For $n=3$ and $p=1$, there exist valuations not
mentioned before.
