  We introduce the notion of Lorentzian fans, which form a special class of
tropical fans that are particularly well-suited for proving Alexandrov-Fenchel
type inequalities. To demonstrate the utility of Lorentzian fans, we prove a
practical characterization of them in terms of their two-dimensional star fans.
We also show that Lorentzian fans are closed under many common tropical fan
operations, and we discuss how the Lorentzian property descends to the
underlying tropical variety, allowing us to deduce Alexandrov-Fenchel type
inequalities in the general setting of tropical intersection theory on tropical
fan varieties.
