This tutorial is probably also available as a Jupyter notebook in the `demo`

folder in the polymake source and on github.

Different versions of this tutorial: latest release, release 4.1, release 4.0, release 3.6, nightly master

**This is an old revision of the document!**

## Semantics of Cones and Polytopes

### General Remarks

The general semantics of a big object in polymake is as follows: a list of properties describes an equivalence class of mathematical objects. Often this equivalence class consists of a single element, but this is not necessary.

All big objects are immutable as mathematical objects. This means it is possible to add more properties, but only consistent ones. Ideally, these properties pre-exist (since they are logically derived from the input description of the object), and the rules only make them explicit.

As an example an object of class Polytope defined by `VERTICES`

gives such a single element equivalence class. A typical example of the other kind is a polytope given combinatorially, in terms of `VERTICES_IN_FACETS`

. An extreme case would be a Polytope object defined by `VOLUME`

only (defining the set of polytopes of all possible dimensions which happen to have that volume). While this makes little sense a similar example would be a Polytope object defined by F_VECTOR only. From this it makes sense to derive, e.g., `N_VERTICES`

or `H_VECTOR`

.

If a user asks for a property which cannot be derived this property is set to undef. This occurs, e.g., if one asks for the `VERTICES`

of a combinatorially defined polytope.

For instance we have four possibilities which can occur for `VERTICES`

. The property

- does not exist (it is not listed in
`list_properties`

): This basically means that the property is not derived/calculated, yet. - exists and is set to
`undef`

: Polymake is not able to derive this property with the given properties. The polytope might be empty or not. - exists and is empty: So the polytope is empty.
- exists and is neither set to
`undef`

nor is empty: Our polytope is not empty and the property returns what you expect.

## Objects of type ''Polytope''

Polytope theory is nice because this is where combinatorics meets metric geometry. For mathematical software dealing with such objects it is necessary to get the semantics straight. Below we describe some pitfalls.

### With coordinates: Geometry

Being non-empty is recorded in the property `FEASIBLE`

. This is `true`

if and only if the polytope is not empty.

A non-empty polytope in R^n is encoded as its homogenization in R^{n+1}. Hence, any non-empty polytope has at least one facet (which may be the far hyperplane [1,0,0,…,0]) and one vertex.

: cdd does not return the facet [1,0,0,…] in redundancy checks. It does so in the dual description. As our model should be completely equivalent in the primal and dual description we will try to use the dual description of cdd for redundancy both in the primal and dual case.

### Without coordinates: Combinatorics

`VERTICES_IN_FACETS`

always describes the combinatorics of a bounded polytope: this is any polytope which is projectively equivalent to the polyhedron defined by `VERTICES`

or `POINTS`

or dually modulo its `LINEALITY_SPACE`

.

Each property must clearly specify if it depends on the geometry or only on the combinatorics.

### Special Cases

Most of what comes below is a consequence of the design decisions explained above.

#### Empty polytopes

With the introduction of the `Cone`

class and redefining `Polytope`

as a derived class (in version 2.9.10) this raises the question of how to deal with empty polytopes.
This is a bit subtle as the cone over an empty polytope does not have a canonical definition. Most text books hence exclude this case. For them a polytope is never empty. There was a time when this was also polymake's point of view (until version 2.3).

However, this was changed for the reason that often people generate systems of inequalities and then look at the feasible region. Most of the time they obtain a polytope and proceed, but sometimes it fails, and the region is empty. It is therefore necessary to give a definition of the empty polytope (geometrically) which is consistent:

An empty polytope is recognized by `FEASIBLE == false`

. Such a polytope is required to have `VERTICES`

and `FACETS`

empty. This is totally different from having `VERTICES`

or `FACETS`

undefined (see above).

#### Zero-dimensional polytopes

A zero-dimensional polytope is a single point. In our model it has one vertex and one facet (the far hyperplane).
`VERTICES_IN_FACETS`

is a 1-by-1 matrix with a zero entry. This means that the single vertex does *not* lie on the single facet.

Such a polytope is both simple and simplicial, i.e. it is a simplex.