Definition 21.8.1E11C7A4F96AC

Definition 21.8.1.E11C7A4F96AC A lattice polytope in $M_{\mathbb R}$ is the convex hull of finitely many points of $M$. A face of a lattice polytope $P$ is a set of the form $\{ m\in P:\langle m,x\rangle =\min _{m'\in P}\langle m',x\rangle \} $ with $x\in N_{\mathbb R}$; faces are nonempty, and $P$ itself is the face for $x=0$. Vertices, edges and facets are the faces of dimension zero, of dimension one and of codimension one, respectively. A nonzero vector of a lattice is primitive if it is not of the form $ky$ with $y$ in the lattice and an integer $k\geq 2$. Let $P$ have nonempty interior. Its vertices are lattice points, each edge is a segment $[v,v']$ between two vertices, and a vertex contained in an edge is one of its endpoints (lemma 21.8.2). Thus an edge $[v,v']$ containing a vertex $v$ determines the primitive edge direction at $v$, the unique primitive vector of $M$ that is a positive multiple of $v'-v$ (see below). The polytope $P$ is smooth, or Delzant, if at each vertex its primitive edge directions form a basis of $M$. In dimension zero the single point is smooth, with the empty basis.