Oswald in the doorway: the blog of the Oswald Innocence Campaign, by Ralph Cinque

Oswald told interrogators that the only thing he brought to work was his lunch in a paper bag, consisting of a cheese sandwich and an apple, which he got from Mrs. Paine's house. He had to get it from there because he and Frazier didn't stop anywhere on their way to work.

Police had a legal and forensic obligation to investigate that claim by looking for the remnants of that lunch in the place that he ate it, the domino room. Presumably, they had a trash can in there.

And, Oswald was still alive at the time, so they could have asked him where he disposed of the remnants of his lunch and gone to retrieve it. 

Those remnants would have included the following:

1) the brown paper bag
2) the sandwich bag (which may have been two because in the Fritz Notes it says cheese sandwiches, plural)
3) possibly bits of crust or any other non-eaten parts of the sandwich or sandwiches
4) the apple core

Police had a duty to retrieve those bits of debris because it meant certain confirmation or refutation of Oswald's lunch claim. If those remnants were found, it meant that he was telling the truth. If they were absent, then it meant that he was lying. And it was pretty much as simple as that. 

So, why was there nothing said about this? Did they or did they not look for the remnants of Oswald's lunch?

I have to think that they did. After all, you didn't have to be Lieutenant Columbo to realize the evidentiary value of it. So, if I were going to bet, I would bet that they did look for it. And, I would also bet that they found it. Why? Because if they didn't find it, it would have been incredibly damning to Oswald. It would have meant that he lied about bringing his lunch in a paper bag. Why would he lie about that? Only because he brought his rifle instead. Isn't that how it would look?  

So, if they looked for those remnants and failed to find them, they would surely have announced it. But, if they did find them, then it meant that he was telling the truth, and they weren't going to announce ANYTHING that supported Oswald's innocence.

So, without knowing for sure, I assert that they looked for and found the remnants of Oswald's lunch.