  Combinatorial mixed valuations associated to translation-invariant valuations
on polytopes are introduced. In contrast to the construction of mixed
valuations via polarization, combinatorial mixed valuations reflect and often
inherit properties of inhomogeneous valuations. In particular, it is shown that
under mild assumptions combinatorial mixed valuations are monotone and hence
nonnegative. For combinatorially positive valuations, this has strong
computational implications. Applied to the discrete volume, the results
generalize and strengthen work of Bihan (2015) on discrete mixed volumes. For
rational polytopes, it is proved that combinatorial mixed monotonicity is
equivalent to monotonicity. Stronger even, a conjecture is substantiated that
combinatorial mixed monotonicity implies the homogeneous monotonicity in the
sense of Bernig--Fu (2011).

Combinatorics;Metric Geometry