I have seen two different ways of describing a Delzant polytope:
From Canna Da Silva https://people.math.ethz.ch/~acannas/Papers/toric.pdf, a Delzant polytope is a polytope in $\mathbb{R}^{n*}$ satisfying the three following properties:
- it is simple: there are exactly $n$ edges meeting at each vertex;
- it is rational: the edges meeting at each vertex are of the form $p + tu_i$, where $t \geq 0$ and $u_i \in \mathbb{Z}^{n*}$;
- it is smooth: for each vertex, the corresponding $u_1,...,u_n$ form a $\mathbb{Z}$-basis of $\mathbb{Z}^{n*}$.
On the other hand, one can look at a Delzant polytope through its half-space description: a Delzant polytope $\Delta$ with $d$ facets can be written as $$ \Delta := \lbrace x \in \mathbb{R}^{n*} \ | \ \langle x, v_j \rangle + a_j \geq 0, \ j=1,...,d \rbrace, $$ where the $v_j$'s are primitive inward-pointing conormals in $\mathbb{Z}^n$, and $a = (a_1,...,a_d) \in \mathbb{R}_{\geq 0}^{d*} \setminus \{0\}$. In this case, https://arxiv.org/pdf/1308.3224.pdf describe the Delzant's properties by:
compactness: each $k$-codimensional face of $\Delta$ is the intersection of exactly $k$ facets;
smoothness: the $k$ associated conormals $(v_{l_1},...,v_{l_k})$ can be extended to a $\mathbb{Z}$-basis for the lattice $\mathbb{Z}^n$.
Although both descriptions seem very similar (at least intuitively), I have trouble proving that they are equivalent. Any help will be greatly appreciated.
Thanks in advance!
