This is a revision of my previous question. The question is motivated by the fact that, for a Linear Program (LP) obtained by relaxing a given Integer Linear Program (ILP), an optimal solution may be very far from those of the given ILP. This can happen if an optimal solution of the LP is a "very sharp" vertex of the LP solution space, with no room for any feasible integer lattice solutions nearby in that space. This ruins the hope of "relaxing ILP to LP, finding an optimal solution $x^{*}$ of the LP, and finding the nearest feasible integer lattice point close to $x^{*}$, hoping it is close-enough-to-optimal for the original ILP". This question is an attempt at formulating the conditions on the "sharpness" of the vertex that will guarantee a near-enough-optimal result of the above approach.
Suppose $P$ is a convex polyhedron in a Euclidean space $\mathbb{E}^{d}, (\cdot, \cdot)$ and contains at least one integer point (to avoid working with trivially empty sets), and $v$ is an extreme point of $P$. Every 1-D edge of $P$ "emanating from $v$" has a director: a unit vector $\hat{n}$ pointing from $v$ along the edge. (E.g., in 3-D, every extreme point of a cube has 3 such incident edges, pairwise perpendicular.) Call such directors, "edge directors at $v$".
The following statement seems true for dimensions $d = 2, 3$:
If the cosine $(\hat{n}_{1}, \hat{n}_{2})$ of every pair of edge directors at $v$ does not exceed some suitable constant $\alpha_{d}$ (the same for all pairs of edge directors)--intuitively, if the angle between every pair of edge directors is obtuse enough,--then $P$ contains an integer point within $l^{1}$-distance 1 from $v$.
For $d=2$, the cosine upper bound $\alpha_{2} = -0.088$ will do the job: only two edge directors emanate from $v$, and at an obtuse angle.
Are there any results that generalize this for higher dimensions $d$?