A generalized cusp $C$ is diffeomorphic to $[0,infty)$ times a closed Euclidean manifold. Geometrically $C$ is the quotient of a properly convex domain by a lattice, $Gamma$, in one of a family of affine groups $G(psi)$, parameterized by a point $psi$ in the (dual closed) Weyl chamber for $SL(n+1,mathbb{R})$, and $Gamma$ determines the cusp up to equivalence. These affine groups correspond to certain fibered geometries, each of which is a bundle over an open simplex with fiber a horoball in hyperbolic space, and the lattices are classified by certain Bieberbach groups plus some auxiliary data. The cusp has finite Busemann measure if and only if $G(psi)$ contains unipotent elements. There is a natural underlying Euclidean structure on $C$ unrelated to the Hilbert metric.
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