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A $d$-dimensional lattice polytope $P$ is Gorenstein if it has a multiple $rP$ that is a reflexive polytope up to translation by a lattice vector. Thedifference $d+1-r$ is called the degree of $P$. We show that a Gorensteinpolytope is a lattice pyramid if its dimension is at least three times itsdegree. This was previously conjectured by Batyrev and Juny. We also present arefined conjecture and prove it for IDP Gorenstein polytopes.

Proof of a conjecture of Batyrev and Juny on Gorenstein polytopes