  We investigate the arithmetic-geometric structure of the lecture hall cone \[
L_n \ := \ \left\{\lambda\in \mathbb{R}^n: \, 0\leq \frac{\lambda_1}{1}\leq
\frac{\lambda_2}{2}\leq \frac{\lambda_3}{3}\leq \cdots \leq
\frac{\lambda_n}{n}\right\} . \] We show that $L_n$ is isomorphic to the cone
over the lattice pyramid of a reflexive simplex whose Ehrhart $h^*$-polynomial
is given by the $(n-1)$st Eulerian polynomial, and prove that lecture hall
cones admit regular, flag, unimodular triangulations. After explicitly
describing the Hilbert basis for $L_n$, we conclude with observations and a
conjecture regarding the structure of unimodular triangulations of $L_n$,
including connections between enumerative and algebraic properties of $L_n$ and
cones over unit cubes.
