Friday, March 17, 2017

Simplex Overlap Equivalnce

What is a polytope? For one thing, it has flat sides. For another, two polytopes can be equivalent without being the same. Anecdotally, all right triangles are in some sense equivalent. I would argue that the various kinds of simplex are limits of sequences of irregular simplices, and all irregular simplices are equivalent up to dimension. I can define polytope as a collection of maximal super-region spaces, together with their intersection super-region spaces. The intersection spaces are necessary because otherwise the relation between the maximals would not be captured. I can convert any finite union of intersections of real vector halfspaces to a polytope, and any sufficiently small tweak of the halfspaces produces the same polytope. Furthermore, my personal criterion for polytope equivalence is satisfied. Simplex overlaps are equivalent iff they are equivalent as polytopes. In other words, a simplex overlap is two maximal super-regions, related by their one intersection. A maximal super-region space is one of a super-region not contained by any other super-region covered by the polytope's regions embedded in the space. A super-region space is a space with all boundaries attached to an inside region. Much as section spaces uniquely identify extensions of a space, super-region spaces uniquely identify super-regions of a space. To find the section space, take the regions attached on one side of the extension boundary. To find the super-region space, take the boundaries attached to the super-region. See below for a proof that there is at most one inside region attached to all boundaries. Thus, a polytope is a collection of super-region spaces together with sidednesses of the super-region wrt attached boundaries. A super-region space is specified as a map from attached boundary to the super-region space of the facet of the boundary. For example, a three dimensional super-region space has a face per attached boundary, each face has a segment per significant boundary, and each segment has a two significant vertex boundaries. A facet is identified by the set of boundaries in the boundary map path that leads to it. Thus, the traditional definition of polytope as a graph of facets suffices for convex polytopes, and for concave polytopes, the definition above suffices.

Super Region Theorem

There is at most one inside region attached to all boundaries. Suppose two points, p and q, are in regions attached to all boundaries. Toss out boundaries without putting either p or q in outside regions, and without putting p and q the same region. Finally every boundary makes one or both of p and q outside or makes p and q the same. If there were a boundary, b, that separates only p from an outside region, then b is not attached to q, so all boundaries except the mutual boundary, c, separate both p and q from outside regions. Consider two outside regions, r and s, opposite p and q wrt boundary b. R and s must be opposite each other wrt c because r and s are the same after removing c. Thus every outside region is attached to c, so there are 6 regions total. This is not possible in any dimension, so there must be at most one region attached to all boundaries.

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