  Classifications of $\rm{SL}(n)$ covariant function-valued valuations are
established with some assumptions of continuity. New valuations, for example,
weighted moment functions, are introduced and our classifications give unified
characterizations of the Laplace transform on convex bodies, $L_p$ moment
bodies, $L_p$ difference bodies, and polar $L_p$ moment bodies ($L_{-p}$
intersection bodies). Using the new classifications, we also establish some
Euler-type relations.

Metric Geometry;Functional Analysis