PD Dr. Andreas Paffenholz

Normality of smooth lattice polytopes

It has been conjectured that all smooth lattice polytopes are projectively normal.

This page contains descriptions of some polytopes that are interesting in connection with this conjecture. The list of examples will be extended. Contact me, if you have questions about these polytopes, of know of an interesting example not listed on this page.

Smooth reflexive polytopes in dimension up to 8 are all normal. You can find them here

There is another a web page on projective normality maintained by Diane Maclagan that contains some references to literature and notes on recent progress.

The polytopes are provided in polymake format. See here for more information on using them.

Lists of Polytopes

Description Data Authors Source
A normal polytope without regular unimodular triangulation. [poly] Ohsugi and Hibi [DCG]

A counterexample to Integer-Caratheodory: The vector [9,13,13,13,13,13] is not generated.

The second polytope is projectively equivalent and bounded.

[poly] [poly] Bruns, Gubeladze, Henk, Martin, Weismantel [Crelle]
Non-normal F4-polytope. [poly] [poly] [poly] [poly] [poly] [poly] Haase and Paffenholz
A polytope that is very ample, but neither normal nor smooth.
The second polytope gives a nicer representation.
[poly][poly] Bruns