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User:MWinter4/Direct sum (polytope theory)

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In polytope theory the direct sum is a binary operation on convex polytopes commonly denoted by or . It is dual to the Cartesian product of polytopes. Like the Cartesian product, the direct sum of two polytopes of dimensions and is a polytope of dimension . The operation behaves well with respect to combinatorial and geometric properties of polytopes.


Geometric construction

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Let be a polytope of dimension and let be a polytope of dimension . Their direct sum can be constructed as follows: we first assume that and are embedded into so that their affine hulls intersect in a single point that lies in the relative interior of both polytopes. The direct sum is then the convex hull of the union . While the geometry of the resulting polytope will depend on the choice of embedding, the combinatorics is independent of this choice.

Instead of chosing an arbitrary embedding, the following standard construction can be applied. We shall assume that both and contain the origin in their respective relative interior. Suppose further that has vertices and has vertices . Then the convex hull of the following points yields a realization of :

and
,

where denotes a list of zeros. Yet another way to write this is

.

The direct sum is dual to the Cartesian product. More precisely, it holds

where denotes the polar dual of and means combinatorial equivalence.


The direct sum can be obtained from the join via projection along the additional dimension.

Combinatorics

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If is a proper face of , and is a proper face of , then the convex hull of is a proper face of the direct sum (where we assume ). The combinatorial type of this face is , where denotes the join of polytopes. For the f-vector of the direct sum is

where and .

Given abstract polytopes and , the direct sum can also be constructed combinatorially as follows. The faces of are pairs , where . The incidence relation is given as follows: ...

If both and are of dimension at least two, then the edge graph of the direct sum is the graph join of the edge graphs of and . In particular, has the same edge graph as the join . This can be use to construct polytopes of different dimensions but with the same edge graph.

Volume

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The volume under the standard construction can be expressed in terms of the volume of and as follows:

...

As a consequence, the Mahler volume of the direct sum can be expressed directly.

Subdirect sum

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Given polytopes and and faces , the subdirect sum

is constructed by embedding and into affine subspaces that intersect only in a point that lies in the relative interior of both and . While the resulting polytope might depend on the choice of , its combinatorics does not.

If one choses , then the subdirect product is the same as the direct product, that is

For this reason one also writes

If the are vertices, then the operation is also called a vertex sum. If one choses as a vertex and as a line segment, then the operation is also called vertex splitting since one replaces the vertex by two vertices, namely, the end vertices of the interval .

Other relations

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  • Hanner polytopes are constructed inductively from line segments by taking Cartesian products and direct sums.

References

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