New York City has one of the lowest violent crime rates "of any big city" for a couple of very basic reasons: 1) it's a police state (nearly 40,000 law enforcement officers, and that's just on the municipal level); and 2) it's a very expensive place to live, which means that most of the city's criminal element has simply moved into neighboring urban/suburban areas. I know for certain that violent crime has escalated dramatically in places like Westchester, Nassau, and Rockland Counties in New York and in much of northern New Jersey.
NYC had 8.1 million people, which is more than twice the population of the next largest city. Lots of people, lots of cops, but not a "police state" by any means. Although there are areas in the Borough of Manhattan that rank with the most expensive places to live in the world, the vast majority of NYC residents live in the the outer boroughs, which are not all that expensive, and in many areas, significantly less expensive than the burbs. I live in the northern burbs, and the violent crime rate is almost non-existant. Many of the "near burbs" are as expensive, if not more expensive than NYC, so if the criminal element can't afford Astoria, Brownsville, and East New York, then they certainly can't afford New Rochelle, White Plains, Huntington, Wantagh, or New City. Unlike NYC, the gun laws in the northern burbs are not that strict. I carry a shotgun in the back of my truck and a legal concealed weapon. If the violent crime rate in the burbs has, in fact, escalated dramatically (and I generally don't think that it has), then I suspect that the reason has nothing to do with the gun laws.