This is no longer a tenable idea.
1) The image above is from a food desert map discussed on Wired, which uses line length and thickness to represent distance from grocery stores. You will note that the thick long red lines in grocery-stores-are-far-away-land are exactly where people are fittest (Rocky Mountain states) and the most grocery stores are in the Piedmont South and Black Belt, exactly where people are most obese. If you're trying to show that food deserts cause people to be healthy, you couldn't do a much better job than this map.
2) As if that's not enough, there has also recently been work (two studies in this article) showing that the idea that low socioeconomic status neighborhoods in American cities are not, after all, food deserts.
The food desert-obesity connection is a myth, and people should stop believing in it.