  Though much is known about ${\bf s}$-lecture hall polytopes, there are still
many unanswered questions. In this paper, we show that ${\bf s}$-lecture hall
polytopes satisfy the integer decomposition property (IDP) in the case of
monotonic ${\bf s}$-sequences. Given restrictions on a monotonic ${\bf
s}$-sequence, we discuss necessary and sufficient conditions for the Fano,
reflexive and Gorenstein properties. Additionally, we give a construction for
producing Gorenstein/IDP lecture hall polytopes.
