In real convex projective geometry, simplices have many characteristics of flat geometry. In particular, we will discuss compact convex projective manifolds (in any dimension greater than 2) which have properly embedded codimension-1 simplices, and demonstrate that examining these submanifolds yields strong topological data about the total manifold. This work generalizes a 2006 theorem of Benoist for compact 3-dimensional convex projective manifolds which gives a JSJ decomposition along properly embedded triangles.