= Polytope
{wiki}
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 $[0,1]^n$ can be described either by its $2^n$ vertices or by the inequalities $0\leq x_i\leq1$. 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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