The 0-1 knapsack problem is known to be NP-complete, and the greedy approach by Dantzig (based on choosing on the basis of density or value/weight) can be shown to be suboptimal using counterexamples. Many of the counterexamples seem to rely on a worst case feature where the weights of the elements are huge and comparable to the capacity of the knapsack.
Suppose in a given instance, this is not so i.e. weights of all elements are very small compared to the capacity of the knapsack. In this case, will the greedy algorithm perform optimally (or atleast near optimally for practical purposes)?
I apologize for such a qualitative description, and am seeking a precise formulation for the question, and a possible solution. 



Greedy approach to 0-1 Knapsack problem in specific instances
I don't think this works.


Suppose you have a counterexample to the greedy heuristic, with a knapsack of size $S$ and all elements of density at most $D$ and weight at most $W$. Now add a large number $N$ of objects of weight $W$ and density $1000D$, and consider a knapsack of size $S+NW$. Clearly the $N$ 'dense' objects are included in the optimal solution, and after adding them you're left with an instance of the previous counterexample. By making $N$ large enough, you can ensure that the weights of all objects are arbitrarily small compared to the size of the knapsack.


