The officials promise more aid as Avenues expands to a second building nearby in 2018, and it has just created a position for someone to work full time on community engagement. “It’s based on our belief that our students will learn more by being part of our community,” said Gardner Dunnan, the academic dean.

To expand Chelsea Market, the food and office complex on Ninth Avenue between 15th and 16th Streets, Jamestown Properties negotiated with the city a pledge of more than $1 million over four years for a technology training program for public housing youth so they can compete for jobs in the area, as well as $12 million for the High Line and about $5 million for a fund to build affordable housing.

But jobs for local residents have not materialized to the extent expected, residents and local officials said. “When you have some of the best known, best paying companies in the United States located in Chelsea,” said Councilman Johnson, “it’d be ideal to try to get young people who are from low-income families to offer paid internships, job training and jobs to get them involved so they could stay in the neighborhood they grew up in.”

More wealth is coming. At the northern edge of Chelsea, the huge new development known as Hudson Yards is rising, with 5,000 luxury apartments, designer stores, new parks and a hotel, continuing the area’s trend toward affluence.

Back in a more modest New York, Ms. Waters gets her hair done at a beauty school on 34th Street for $15, and alternates among three restaurants she can afford for the occasional meal out. As the vice-president of the tenants’ association at Elliot Houses and president of the Hudson Guild’s neighborhood advisory committee, she is now working with other residents to rent a van or a bus in November for a holiday shopping trip to New Jersey. In between cross-state trips, she studies shoppers and saves coupons before she hunts for deals closer to home.

But on balance, she said, “I’d rather have Chelsea as it is today.”

“There’s more people,” she said. “It’s brighter, it’s beautiful, it’s more inviting than it used to be. We’re very lucky to be able to stay in housing that hopefully will not disappear.”