Self Dual Reflexive Simplices with Eulerian Polynomials

lattice polytope 𝒫 is called reflexive if its dual 𝒫^∨ is a lattice polytope. The property that 𝒫 is unimodularly equivalent to 𝒫^∨ does not hold in general, and in fact there are few examples of such polytopes. In this note, we introduce a new reflexive simplex Q_n which has this property. Additionally, we show that δ -polynomial of Q_n is the Eulerian polynomial and show the existence of a regular, flag, unimodular triangulation.