11 Facts About Convex polytope

Convex polytope is a special case of a polytope, having the additional property that it is a convex set contained in the -dimensional Euclidean space.

An open convex polytope is defined in the same way, with strict inequalities used in the formulas instead of the non-strict ones.

Fortunately, Caratheodory's theorem guarantees that every vector in the Convex polytope can be represented by at most d+1 defining vectors, where d is the dimension of the space.

Theodore Motzkin proved that any unbounded polytope can be represented as a sum of a bounded polytope and a convex polyhedral cone.

In other words, every vector in an unbounded polytope is a convex sum of its vertices, plus a conical sum of the Euclidean vectors of its infinite edges .

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Every convex polytope is the image of a simplex, as every point is a convex combination of the vertices.

Face of a convex polytope is any intersection of the polytope with a halfspace such that none of the interior points of the polytope lie on the boundary of the halfspace.

The whole Convex polytope is the unique maximum element of the lattice, and the empty set, considered to be a -dimensional face of every Convex polytope, is the unique minimum element of the lattice.

Convex polytope can be decomposed into a simplicial complex, or union of simplices, satisfying certain properties.

Different representations of a convex polytope have different utility, therefore the construction of one representation given another one is an important problem.

Task of computing the volume of a convex polytope has been studied in the field of computational geometry.