The following article contains an answer for the question: What fact about Casanova gives Alex Cross hope that there is time to rescue the imprisoned women? , can you please find it? 

Washington, D.C. detective and forensic psychologist Alex Cross heads to Durham, North Carolina when his niece Naomi, a college student, is reported missing. He learns from police detective Nick Ruskin that Naomi is the latest in a series of young women who have vanished. Soon after his arrival, one of the missing women is found dead, bound to a tree, and a short time later, Kate McTiernan is kidnapped from her home.
When she awakens from a drugged state, Kate discovers that she is being held by a masked man calling himself Casanova, and she is one of several prisoners trapped in his lair. She manages to escape and is severely injured when she jumps from a cliff into a river. After she recuperates, she joins forces with Cross to track down her captor, whom Cross concludes is a collector, not a killer, unless his victims fail to follow his rules. This means there is time to rescue the other imprisoned women, as long as they remain obedient.
Clues lead them to Los Angeles, where a series of gruesome kidnappings and murders have been credited to Dr. William Rudolph, known as the Gentleman Caller. Cross's efforts to capture and question Rudolph are foiled when Rudolph escapes. In North Carolina, Cross traces Casanova up the river. Alerted by a gunshot, he discovers Casanova's underground hideout. Rudolph is revealed to be Casanova's partner. Casanova escapes, while Rudolph is shot by Cross. Cross rescues the kidnapped women, including Naomi.

a collector, not a killer