A convex polytope is a bounded intersection of finitely many closed half-spaces, equivalently the convex hull of finitely many points.
A vertex of a convex polytope is an extreme point: it cannot be expressed as a nontrivial convex combination of two distinct points in that polytope. In a full-dimensional simple polytope of dimension , a vertex lies on exactly incident facets.
A facet of a full-dimensional convex polytope in is a face of dimension . Each irredundant supporting closed half-space defines one facet.
If and a convex polytope satisfies , then it has at least facets. Its outward facet normals define spherical caps covering the unit sphere; the spherical cap area upper bound supplies the estimate.
The convex hull of the positive and negative coordinate unit vectors. Its boundary triangulates a sphere, and each face contains at most one member of each antipodal pair of coordinate unit vectors. This makes its barycentric subdivision useful for triangulating Real projective space.
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A **convex polytope** is a mathematical object that generalizes the concept of polygons and polyhedra to higher dimensions. More formally, a convex polytope can be defined in several ways, including: 1. **Geometrically:** A convex polytope is a bounded subset of Euclidean space that is convex, meaning that for any two points within the polytope, the line segment connecting them is also contained within the polytope.
