On a warm afternoon in Liberty City, three women stood behind a table offering a smattering of fresh produce: 10 tomatoes, five cucumbers, three peppers, five mangoes, one cabbage and a large box of green beans.

They represented a nonprofit, Urban GreenWorks, dedicated to bringing fresh produce to poor neighborhoods with the idea that better food choices will help reverse the wave of obesity sweeping America, particularly devastating in low-income areas.

Though their table was set up on the shaded sidewalk on busy Northwest 62nd Street, not a single customer approached the stand over a 30-minute period.

Meanwhile, two blocks away at the Monar Market, Ronnie Othman was ringing up the cash register about every 30 seconds: Fritos and a Coke, a packaged cookie and Seven Up, Doritos and a Coke. The convenience store offered some apples and bananas in a basket, and a lone head of lettuce in a refrigerated case. “But they don’t sell all the time,” Othman said.

Studies by the Centers for Disease Control show that all of America is getting fatter -- rich and poor, all ethnic groups, all education levels. Still, the poor are more likely to be obese than the affluent. First Lady Michelle Obama and many nutritional activists believe that one reason the poor tend to be fat is that they often live in “food deserts,” places without large supermarkets offering fresh fruits and vegetables available in more affluent neighborhoods.

That theory is questioned by many researchers. “The idea of food deserts has nothing to do with obesity,” says Roland Sturm, a Rand Institute researcher who co-authored a recent paper on a study of 13,000 children that found no correlation between kids’ obesity and access to fresh produce. A much better predictor, he says: parents’ weight.

“You’re just not going to change behavior by offering more vegetables,” said Sturm in a telephone interview. “You can see people who shop at Whole Foods are thinner, but that doesn’t mean that if you had a Whole Foods on every corner, everybody would be thinner.”

Sturm’s work is backed up by a 2011 study in the Archives of Internal Medicine, which followed 5,100 people over 15 years and found that greater supermarket availability had no correlation with how frequently people ate fruits and vegetables.

Still, major efforts nationally and locally are focused on making healthy foods available to the poor. The Obama administration in 2009 targeted $370 million in stimulus funds for a program called Communities Putting Prevention to Work, with much of the money intended to fight obesity. About $14.7 million of the money went to Miami-Dade County to reduce obesity through initiatives that included getting more nutritional food into public schools and creating bike paths to promote exercise.

The Miami-Dade Health Department received $2.7 million to fund 14 initiatives, including hiring a dozen employees, sponsoring wellness seminars and funding a 16-page report, released in March, called “Expanding Supermarket Access in Areas of Need,” with a map showing large sections of northwestern Miami-Dade — particularly parts of Liberty City, Opa-locka and Hialeah — as “areas of greatest need.”

Health Department Administrator Lillian Rivera wrote in the report that “research shows that better access to affordable, nutritious food is associated with healthier eating habits.’’ She noted that “250,000 Miami-Dade residents (10 percent) live in low-income areas that have poor supermarket access and higher than average death rates from diet-related causes.”