That's from a new paper by Hunt Allcott, Rebecca Diamond, Jean-Pierre Dubé. This is one of the most rigorous investigations I've seen of the causal impacts of the "food environment" (in this case, the presence of grocery stores and movements of people into "healthier" neighborhoods) on dietary choice.
What did they find? From the conclusions:
The authors end by concluding that policy efforts to alter local food supplies are likely to be ineffective. Their data strongly supports this conclusion. They recommend, instead, to use public policy to improve health education. I'm surprised they make this recommendation because their study provides no indication that more education would be a cost-effective intervention. If anything, what their study shows is that economic development (turning low-income households into high-income households) is the most effective way to improve the healthiness of dietary choice.
Hat tip to Alex Tabarrock at the Marginal Revolution blog who is highly skeptical of the food desert concept.