The functional form of the dual mixed volume

This paper is to generalize the dual mixed volume of star bodies to that of dual quasi-concave functions by extending the radial Minkowski linear combination of star bodies to that of dual quasi-concave functions. We attempt to build up some functional versions of notions and inequalities from the dual Brunn-Minkowski theory. In particular, both the dual mixed Brunn-Minkowski inequality of star bodies and the dual Aleksandrov Fenchel inequality of star bodies are generalized to that of dual quasi-concave functions.