A convex polytope is the convex hull of finitely many points in a finite-dimensional real vector space. Equivalently, it is a bounded intersection of finitely many closed half-spaces. Its affine dimension may be smaller than that of the ambient space. Broader uses of polytope, including abstract or nonconvex versions, must specify their convention.
The simplex is the convex hull of affinely independent points; a square is a two-dimensional polytope with four vertices. The cube can be described either by its vertices or by the inequalities . These are its vertex and half-space descriptions.
A linear objective on a nonempty polytope attains its maximum at some vertex: every point is a convex combination of vertices, and its objective value is the same combination of vertex values. This links the geometry to linear programming. The maximizing vertex need not be unique, and an entire face may attain the optimum.
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A polytope is a geometric object with "flat" sides, which exists in any number of dimensions. The term is commonly used in the contexts of both geometry and higher-dimensional mathematics. Here are some key points about polytopes: 1. **Definition**: A polytope is defined as the convex hull of a finite set of points in a Euclidean space. Essentially, it is the shape formed by connecting these points with flat surfaces.
