$\mathrm{SL}(n)$ contravariant function-valued valuations on polytopes

We present a complete classification of $\mathrm{SL}(n)$ contravariant, $C({\mathbb{R}}^n\setminus \{o\})$-valued valuations on polytopes, without any additional assumptions.It extends the previous results of the second author [Int. Math. Res. Not. 2020] which have a good connection with the $L_p$ and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of $\mathrm{SL}(n)$ contravariant symmetric-tensor-valued valuations on polytopes.