Examples and Counterexamples to some Conjectures
Contact me if you have questions about these polytopes or suggestions for polytopes that should be studied.
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 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.