In the case of a food desert within a residential area, it's probably a lot easier to boost demand by artificially increasing supply, rather than the other way around.
> 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.
On the topic more generally
> Dr. Sturm found no relationship between what type of food students said they ate, what they weighed, and the type of food within a mile and a half of their homes.
https://www.nytimes.com/2012/04/18/health/research/pairing-o...
Original study
> School and Residential Neighborhood Food Environment and Diet Among California Youth
> No robust relationship between food environment and consumption is found. A few significant results are sensitive to small modeling changes and more likely to reflect chance than true relationships.
https://www.ajpmonline.org/article/S0749-3797%2811%2900849-X...
> THE GEOGRAPHY OF POVERTY AND NUTRITION: FOOD DESERTS AND FOOD CHOICES ACROSS THE UNITED STATES
> 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.