Examples and Counterexamples to some Conjectures
This page contains examples and counter-examples for various problems. For most constructions I used polymake, some were found using the lists of smooth reflexive polytopes.
Contact me if you have questions about these polytopes or suggestions for polytopes that should be studied.
Polytopes can directly be loaded into polymake. Try the online version if you don't yet have it. Tarballs can be read into polymake using the tarball-script described here.
Polyhedral Adjunction
For background on these examples see arxiv:1105:2515.
For a polytope P defined by a reduced primitve system Ax≤b the adjoint polytope at height c is civen by the system Ax+c1≤ b. For smooth polytopes P in dimensions 2 and 3 with interior lattice points the adjoint at height 1 is still a lattice polytope. This is false starting from dimension 4.Symmetric Kähler-Einstein Manifolds
For background on these examples see arxiv:0905.2054.
| Dimension 7: | [poly] |
| DImension 8: | [poly] [poly] |
Star Ewald Conjecture
For background on these examples see arxiv:0904.1686.
| Dimension 6: | [poly] [poly] [poly] |
| Dimension 7: | [tgz] (35 polytopes) |
Special Realisations of 3-Polytopes and their Duals
To my knowledge, this problem was posed by Grünbaum and Shepherd in 1997, but probably it is much older. The question is wether you can, for a given 3-polytope P, find a geometric realisation such that the dual polytope has a realisation in which the vertices lie on the facets of P. As far as I know, this is still open, although the general belief seems to be that this is not possible for all 3-polytopes.
Of course, there are some obvious examples, like the cube, or the simplex. Some time ago I found the following three less obvious examples:
- Truncated cube [ poly jvx eps view ]
- Truncated cross polytope [ poly jvx eps view ]
- Truncated tetrahedron [ poly jvx eps view ]
In this fashion I can construct many more examples, but I don't see a general approach. I would be very much interested in new ideas on this question.