The brutal attack in December sparked public debate and fury over chronic sexual violence faced by women in India.
"I am challenging this verdict," said A P Singh, who has defended all four men at various times.
The High Court in New Delhi said it would begin hearing prosecution arguments today, while defence solicitors file their appeals against the convictions and sentences.
"We have to deal with this as expeditiously as possible because the sword of death is hanging over them," Judge Pratibha Rani said.
It could take weeks or months for the court to hear arguments, review evidence and consider the appeals before deciding whether to confirm the execution orders. A review is required for all death penalty cases in India, and the same court is hearing the men's appeals.
In sentencing the four, trial court Judge Yogesh Khanna said the crime had "shocked the collective conscience" of India.
The four had been joyriding in New Delhi on a bus on the night of December 16 when they lured the 23-year-old woman and her male friend into boarding. They beat the friend, took turns raping the woman and abusing her. She died from internal injuries two weeks later.
Another defendant hanged himself in prison, though his family insists he was killed. And an 18-year-old who was a juvenile at the time of the attack was sentenced in August to a maximum of three years in a reform home.