Affine Function-Valued Valuations. (3rd October 2018) - Li, Jin
Abstract: A classification of $\mathrm{SL}(n)$ contravariant, continuous function-valued valuations on convex bodies is established. Such valuations are natural extensions of $\mathrm{SL}(n)$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn–Minkowski theory, for $p \geq 1$ . Hence, our result will help to better understand extensions of the $L_p$ Brunn–Minkowski theory. In fact, our results characterize general projection functions, which extend $L_p$ projection functions ($p$ -th powers of the support functions of $L_p$ projection bodies) to projection functions in the $L_p$ Brunn–Minkowski theory for $0< p < 1$ and in the Orlicz Brunn–Minkowski theory.