If we really want to encourage those in food deserts to eat healthy, we need to provide easy access for them to obtain fresh produce. Lets use the approach of delivering of fresh produce the same way as the ice cream truck used to drive in our neighborhoods and sell ice cream. The truck can stop in front of houses or street corners and sell or give the veggies/ fruit away. They could even use that nice chime or ring as the ice cream truck to let the neighborhood know when they arrive.
Why not incorporate our local prisons and jails to grow the veggies on their vacant land and deliver to the same areas that they may have grown up in and committed crime. Give the inmates a chance to give back to their community. This way we could grow the veggies with essentially free labor and possible give the produce away for a lot less than other processed food or fast food. The inmates will gain valuable farming skills they could use to obtain employment when let out and more importantly gain a sense of self worth that they are giving back to their community. I've seen jails in Wisconsin grow veggies for the local food bank but lets take it to the next step and deliver the harvest straight to the food deserts that are in just as desperate need for fresh healthy produce.