## Monday, December 10, 2012

### Polytopes Convexity Collinearity

Wikipedia defines topological polytope as a graph. A graph is not sufficient to represent simplex overlaps. To represent simplex overlaps I need a graph together with sidedness with respect to boundaries. Sidedness captures convexity and collinearity; otherwise overlaps would be equivalent to polygons. A polytope with sidedness can be represented by a nonlinear sidedness space. Start with a linear space and choose regions in the polytope. Choose a partition of the chosen regions. If the partition is well behaved, a nonlinear sidedness space, with the partition as regions and the same boundaries as the starting linear space, is well defined. The following is a nonlinear space defined by the partition (A, B, C, D) of the boundaries (0, 1, 2, 3, 4, 5).

The partitions are bounded by the black lines, and the polytope is the union of all partitions except D. In the nonlinear space representing the polytope, the following sidednesses hold.

A<-0->B (region A is opposite region B with respect to boundary 0)
A<-0->D
A<-1->B
A<-1->D
A<-2->B
A<-2->D
A<-3->C
A<-3->D
A<-4->D
A<-5->D
B<-0->C
B<-1->C
B<-2->C
B<-3->D
B<-4->D
B<-5->D
C<-1->D
C<-2->D

The following is a poorly behaved partition that does not define a unique sidedness space representation.

This poorly behaved partition suggests the following sidednesses. Note for example, that A and B are opposite each other with respect to 2, and both are opposite C with respect to 2, a contradiction.

A<-0->C
A<-0->D
A<-1->C
A<-1->D
A<-2->B
A<-2->C
A<-2->D
A<-3->C
A<-3->D
B<-0->C
B<-0->D
B<-1->C
B<-1->D
B<-2->C
B<-2->D
B<-3->C
B<-3->D
C<-1->D
C<-3->D
C<-4->D

I propose that nonlinear sidedness space representations of complicated polytopes can be built up from such representations of simpler polytopes. Thus, overlap polytopes are built up from pairings of simplex polytopes embedded in a linear space.

First define well behaved sidedness polytope as a partition of regions from a linear space that has a unique sidedness space representation. Then define the overlap of two sidedness polytopes as their refinement. Then prove the following theorem.

Polytope Theorem: The overlap between well behaved sidedness polytopes is well behaved.