Convex core

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When the facets of a polytope intersect one another, the facet hyperplanes may partition off a hypervolume at the center of the polytope in the shape of another polytope. This second polytope is the convex core, or kernel, of the original polytope. Particularly complicated polytopes may have many such partitions in them, but only the one at the center is the convex core.

For example, the convex core of a small stellated dodecahedron is a dodecahedron. Conveniently, its edges can be seen without having to dive into the bulk of the polytope, but this is not always the case.

The convex core of a convex polytope is the same polytope. Hence the concept is usually only applied to self-intersecting polytopes.

The convex core has at least the same symmetry as the original polytope. If the original polytope is not degenerate, it has the same rank.

The convex core depends on the choice of center. For a sufficiently symmetric polytope, the center of symmetry is usually implicitly chosen.

The dual operation to the convex core is the convex hull. In other words, the convex core of a polytope is the dual of the convex hull of the dual polytope, and vice versa.

The convex core is not always a polytope; in general it is a convex polyhedron (meaning an intersection of finitely many hyperplanes, unlike a polytope it is not necessarily bounded). In the case where it is not bounded, the dual-hull-dual construction produces a non-convex polytope.

When facets go through the center of the polytope, such as in the octahemioctahedron, it has no convex core.