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In the geometry of numbers one of the main objects of study are lattice-point-free convex bodies. In this talk I will give a survey of conjectures and results on the corresponding objects in the world of lattice polytopes: lattice polytopes without interior lattice points. It turns out that there is a nice invariant in Ehrhart theory, called the codegree, which measures the 'hollowness' of these lattice polytopes. In joint work with Christian Haase and Sam Payne, we could prove a structure theorem on lattice polytopes with large codegree, which yields a new finiteness result on Ehrhart polynomials. If time allows, I will also present related work in progress with Sandra Di Rocco and Christian Haase.