Sharp volume and multiplicity bounds for Fano simplices

We present sharp upper bounds on the volume, Mahler volume and multiplicity for Fano simplices depending on the dimension and Gorenstein index. These bounds rely on the interplay between lattice simplices and unit fraction partitions. Moreover, we present an efficient procedure for explicitly classifying Fano simplicies of any dimension and Gorenstein index and we carry out the classification up to dimension four for various Gorenstein indices.