Access isn't the issue.

I have posted critically about food deserts earlier. The notion is that obesity in low income neighborhoods is due to an absence of good food procurement choices. This was a conjured rationale to support a feel-good initiative out of Washington, D.C. as documented in those earlier posts. My criticism is that we are wasting time and money on nonsense initiatives which serve to allow some people to feel good about their virtue without actually making anything better for the nominal beneficiaries.

Adding a nail in this pathetic coffin is The Geography of Poverty and Nutrition: Food Deserts and Food Choices Across the United States by Hunt Allcott, Rebecca Diamond, and Jean-Pierre Dubé. Abstract:

We study the causes of “nutritional inequality”: why the wealthy tend to eat more healthfully than the poor in the U.S. Using two event study designs exploiting entry of new supermarkets and households' moves to healthier neighborhoods, we reject that neighborhood environments have economically meaningful effects on healthy eating. Using a structural demand model, we find that exposing low-income households to the same food availability and prices experienced by high-income households would reduce nutritional inequality by only 9%, while the remaining 91% is driven by differences in demand. In turn, these income-related demand differences are partially explained by education, nutrition knowledge, and regional preferences. These findings contrast with discussions of nutritional inequality that emphasize supply-side issues such as food deserts.