  In this paper we study the novel notion of thin polytopes: lattice polytopes
whose local $h^*$-polynomials vanish. The local $h^*$-polynomial is an
important invariant in modern Ehrhart theory. Its definition goes back to
Stanley with fundamental results achieved by Karu, Borisov & Mavlyutov,
Schepers, and Katz & Stapledon. The study of thin simplices was originally
proposed by Gelfand, Kapranov and Zelevinsky, where in this case the local
$h^*$-polynomial simply equals its so-called box polynomial. Our main results
are the complete classification of thin polytopes up to dimension 3 and the
characterization of thinness for Gorenstein polytopes. The paper also includes
an introduction to the local $h^*$-polynomial with a survey of previous
results.
