### Defining polyhedral complexes

Posted:

**07 Sep 2017, 14:26**Dear developers,

in my research I often stumble upon collections of combinatorially defined polyhedra, presented as intersection of half-planes, which I suspect constitute a polyhedral complex.

I would like to check whether that is the case. However, according to the documentation,in polymake the input of a polyhedral complex consists of a point configuration together with a collection of point subsets corresponding to vertices of the cells in the complex. In particular, this requires knowledge of the vertex-maximal cell incidences, which is not readily available if all you have are inequality descriptions for your polyhedra.

Would it be possible to relax the input of a polyhedral complex, so that it can be defined by a collection of polyhedra (without prior knowledge of vertices/rays)? The idea would be that polymake produces the vertex/maximal cell incidences, and from that knowledge can test if the collection indeed corresponds to a polyhedral complex, whether it corresponds to a regular subdivision of a polyhedron, whether it is regular, etc.

I am aware that this is not the most general way to construct a polyhedral complex, because it somehow neglects any underlying point configuration on top of which the complex may be defined, but it would be convenient to have it implemented in case you only have inequality descriptions for the cells at hand.

in my research I often stumble upon collections of combinatorially defined polyhedra, presented as intersection of half-planes, which I suspect constitute a polyhedral complex.

I would like to check whether that is the case. However, according to the documentation,in polymake the input of a polyhedral complex consists of a point configuration together with a collection of point subsets corresponding to vertices of the cells in the complex. In particular, this requires knowledge of the vertex-maximal cell incidences, which is not readily available if all you have are inequality descriptions for your polyhedra.

Would it be possible to relax the input of a polyhedral complex, so that it can be defined by a collection of polyhedra (without prior knowledge of vertices/rays)? The idea would be that polymake produces the vertex/maximal cell incidences, and from that knowledge can test if the collection indeed corresponds to a polyhedral complex, whether it corresponds to a regular subdivision of a polyhedron, whether it is regular, etc.

I am aware that this is not the most general way to construct a polyhedral complex, because it somehow neglects any underlying point configuration on top of which the complex may be defined, but it would be convenient to have it implemented in case you only have inequality descriptions for the cells at hand.