Faced with unprecedented overcrowding in New York City’s homeless shelters, Mayor Michael R. Bloomberg has cut through the bureaucracy to open up new shelters across the city, pushing some into wealthy areas where residents fear their new neighbors will bring crime and drugs. Mr. Bloomberg has opened 10 new homeless shelters in recent months in response to the escalating crisis. At last count, more than 46,000 people sought shelter every night in New York, the highest number ever recorded. A recent census report found the city harbors a disproportionate 14 percent of the nation’s homeless, with Los Angeles a distant second at 3 percent. The crisis stems from a lack of affordable housing and the city’s ever-widening gap between the rich and the poor, one of the widest in the U.S. and comparable to that found in sub-Saharan Africa. Unlike in other cities, New York is required by law to provide shelter to every person who seeks it. As the crisis escalates, the city has used its power of emergency authority to create shelters as quickly as possible, bypassing the typical community approval process. And that has created turmoil in neighborhoods like the Upper West Side, where a shelter opened in August among apartments occupied by the wealthiest 1 percent of Americans.