Lipschitz class, narrow class, and counting lattice points

A well-known principle says that the number of lattice points in a bounded subset $S$ of Euclidean space is about the ratio of the volume and the lattice determinant, subject to some relatively mild conditions on $S$. In the literature one finds two different types of such conditions: one asserts the Lipschitz parameterizability of the boundary $\partial S$, and the other one is based on intersection properties of lines with $S$ and its projections to linear subspaces. We compare these conditions and address a question, which we answer in some special cases. Then we give some simple upper bounds on the number of lattice points in a convex set, and finally, we apply these results to obtain estimates for the number of rational points of bounded height on certain projective varieties.