Lp Minkowski valuations on polytopes

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