I wanted to update on 3 small new results.

First, Luis Ferroni used LLM (and a lot of persistence and cleverness) to prove something I long suspected: that IDP lattice polytopes do NOT have unimodal $h^\ast$ vectors. Unimodality is nice because it makes the sequence especially simple: it rises, then falls again… a dromedary, instead of a camel. And that IDP polytopes (that is, polytopes which are generated by atoms on the same level, that is, level one) satisfy it was a (I thought unlikely, but long standing) conjecture of Stanley….

Luis found, quite amazingly, that these are rather nice polytopes: smooth Cayley polytopes of rectangular prisms. This complements our work on IDP polytopes, proving among other things monotonicity of the $h^\ast$ vector in the second half. Congratulations Luis!

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