Editing Polytope (section)

===Convex polytopes===
{{Main|Convex polytope}}
A polytope may be ''convex''. The convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is sometimes defined as the intersection of a set of [[half-space (geometry)|half-spaces]]. This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., in [[linear programming]]. A polytope is ''bounded'' if there is a ball of finite radius that contains it. A polytope is said to be ''pointed'' if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set <math>\{(x,y) \in \mathbb{R}^2 \mid x \geq 0\}</math>. A polytope is ''finite'' if it is defined in terms of a finite number of objects, e.g., as an intersection of a finite number of half-planes.
It is an [[integral polytope]] if all of its vertices have integer coordinates.

A certain class of convex polytopes are ''reflexive'' polytopes. An integral {{nobr|<math>d</math>-polytope}} <math>\mathcal{P}</math> is reflexive if for some [[integer matrix|integral matrix]] <math>\mathbf{A}</math>, <math>\mathcal{P} = \{\mathbf{x} \in \mathbb{R}^d : \mathbf{Ax} \leq \mathbf{1}\}</math>, where <math>\mathbf{1}</math> denotes a vector of all ones, and the inequality is component-wise. It follows from this definition that <math>\mathcal{P}</math> is reflexive if and only if <math>(t+1)\mathcal{P}^\circ \cap \mathbb{Z}^d = t\mathcal{P} \cap \mathbb{Z}^d</math> for all <math>t \in \mathbb{Z}_{\geq 0}</math>. In other words, a {{nobr|<math>(t + 1)</math>-dilate}} of <math>\mathcal{P}</math> differs, in terms of integer lattice points, from a {{nobr|<math>t</math>-dilate}} of <math>\mathcal{P}</math> only by lattice points gained on the boundary. Equivalently, <math>\mathcal{P}</math> is reflexive if and only if its [[dual polyhedron|dual polytope]] <math>\mathcal{P}^*</math> is an integral polytope.<ref>Beck, Matthias; Robins, Sinai (2007), ''[[Computing the Continuous Discretely|Computing the Continuous Discretely: Integer-point enumeration in polyhedra]]'', Undergraduate Texts in Mathematics, New York: Springer-Verlag, {{ISBN|978-0-387-29139-0}}, MR 2271992</ref>