Almost nobody has a weight problem in West Harlem.

Or at least they’re not supposed to, because Fairway is smack in the heart of it, selling fresh produce at decent prices and even offering a free shuttle service for neighborhood residents. It’s been there for over 15 years — that is, a generation of people have grown up alongside it.

But obesity is much more prevalent in West Harlem than in Greenwich Village. This is a problem for the idea that “food deserts” make poor people disproportionately overweight.

The story goes that supermarkets with low-priced fresh vegetables and fruit moved out of poor neighborhoods amidst white flight. This, we are told, left residents with paltry and overpriced produce from bodegas. The result: salty, fattening fast food and junk food as the only viable alternatives.

The idea is now so entrenched that undergraduates often cite it earnestly. It’s on the tonguetips of people at any Blue America dinner party. It sticks easily in the memory and even feels good, because it entails that the obesity problem is due in part to racism. The institutional kind, mind you — maybe call it injustice.

And all of us, laudably, want to call attention to injustice, especially if we are lucky enough not to suffer from much of it.

Hence the food desert idea is now common wisdom. Yet it’s impossible to live in New York and not suspect that something doesn’t quite work about this thesis.

There are far too many C-Towns and Met Foods in humbler neighborhoods with low-priced produce spilling out of the bins, where, nevertheless, upper East Side-style svelteness is not exactly epidemic.

One may well see this but process each one of the supermarkets in poor neighborhoods as exceptions that somehow still preserve the rule. But the cracks in the plaster have always been more than anecdotal. The U.S. Department of Agriculture’s food desert locator, for example — unveiled in 2011 — found almost no food deserts in New York City except in some of the wide-open spaces near Kennedy airport.

Yet for a while, academic studies only had so much to tell us about whether there was a correlation between waistline size and how far away the supermarket is. But these days, the data is in.

As far back as 2006, there was sunny coverage in these parts of the Healthy Bodegas Initiative, stocking bodegas in Harlem, the Bronx and Brooklyn with fresh produce and lowfat milk. The media have been less interested in the uninspiring outcome. By 2010, people were buying more vegetables in only one in four bodegas, for example.

Also last year, a major study under the Nutrition Transition Program led by Dr. Barry Popkin showed that proximity of supermarkets has not affected people’s eating habits, in several cities, over 15 years. Again, there was a mere flurry of coverage.

Of late, the evidence is becoming crushing that the emperor has no clothes. Helen Lee’s study at the Public Policy Institute now confirms the impression that a walk around New York City suggests: Nationwide, the neighborhoods with lots of bodegas and fast food joints also tend to be the ones with the most supermarkets.

Lee documents a Camden, N.J., neighborhood where residents, unaware of the “food desert” notion assumed across the river at Philadelphia cocktail parties, casually say that produce is readily available. A Pathmark and a Save-a-Lot sit right among McDonald’s and Burger King franchises.

Roland Sturm at RAND studied 13,000 California children and then middle school students nationwide. He found, both times, that a supermarket close by doesn’t make a kid thin and living far from one doesn’t make him fat.

Michelle Obama’s Healthy Food Initiative is well intended. But her claim that “if people want to buy a head of lettuce” they have to “take two or three buses, maybe pay for a taxicab, in order to do it,” just doesn’t jibe with the facts.

Food carts, which have proliferated in New York City lately, are a good thing, of course. But so is flossing. So is balancing your tires. The key point is that supermarkets have never been inaccessible to poor people in the way that we have been told.

One lesson is surely plain to everyone on some level: How you eat is due as much to cultural preferences as to how far away a supermarket is.

For black people, for instance, the problem is more a matter of history than where the Key Food is. Slavery and sharecropping didn’t make healthy eating easy for black people back in the day. Salt and grease were often what they were stuck with. Southern blacks brought their culinary tastes north. Zora Neale Hurston used to bless her friend Langston Hughes with fried chicken dinners.

It’s what soul food is, and it’s unclear to me that anyone would deny its centrality to black culture. If I am at an event where one of the main reception snacks is fried chicken drummies, it is almost certainly a black one. The person who makes collard greens with hamhocks is usually not white.

In fact, sometimes the person is me, as I myself was raised on the soul food palate to an extent. I will spend my life resisting that taste for grease; in my heart of hearts, I’ll take Cheetos over baby carrots any day.

We need to give this cultural preference a tune-up: books like Wilbert Jones’ “Healthy Soul Food Cookbook” are great ways to, as it were, have your cake and eat it too.

Then, the proliferation of fast food outlets in poor neighborhoods and the array of cheap junk food in every corner store has as much to do with our problem as how far away the supermarket is. Such food tempts palates raised on too much salt, grease and processed sugar, and parents are often unaware of how unhealthy it is.

For example, another guilty pleasure of mine is Jolly Rancher Soda. Yes, it exists, and it might as well be the hard candy itself me-lted down to a goo and carbonated. I allow myself one every four months — but only find it in corner stores in poorer neighborhoods.

Another thing to target is what people consider a schlep to be. One typical “food desert” piece interviews a woman of 50 who considers 12 blocks a daunting distance to the supermarket. But I’m 46 and that’s how far the nearest supermarket is from me. I think of it as a handy way to get some exercise and fresh air.

Are we who work out and pride ourselves as walking New Yorkers doing poor people a favor by calling it racism when the supermarket is a 15 minute walk down the road?

Yet a possible response will be, “Of course, but we still have to think about how far the Met Foods is from people’s houses.” To many, that will feel more important, more urgent to share, than the part about changing people’s habits.

But the studies are clearly showing that we have to put on a different pair of glasses for this issue. Sure, we must be always wary of the system holding people down. But we must do this with the welfare of the people in mind, not as a way of making ourselves feel good about our own enlightenment.

Poor people do have access to healthy food: This is good news. If anyone finds it unwelcome, inconvenient or even just unengaging, we must question their motives.

jmcwhort@gmail.com