The Naked Sun

The Naked Sun is a 1957 Science Fiction and Crime Fiction Novel by Isaac Asimov, the second of his "Robot Trilogy".

Elijah "Lije" Baley, freshly promoted from his previous successful case in The Caves of Steel, is contracted by political machinations far above his pay grade to investigate a murder on one of the "Spacer" Worlds - Solaria, the richest, most sparsely populated, and most technologically advanced of them all. This makes it, of course, the absolute opposite of the overpopulated, crowded, technologically backward Earth of his exclusive experience. On Solaria, humans are thoroughly outnumbered by their robot servants, to the point where every individual person has an estate of hundreds or thousands of acres, maintained entirely by robot, on which they are the only inhabitant, with the possible exception of a spouse. Even in cases of marriage, Solarians consider it all but inconceivable to come into presence of another human being for any purpose, though Delmarre did so as a matter of "duty". Along with his agoraphobia, this is one of the major barriers to Lije's investigation.

The victim is Rikaine Delmarre, a well-respected but not well-liked native Solarian. The only suspect is his beautiful wife, Gladia, who maintains her innocence despite having been the only person within hundreds of miles at the time of the murder. Lije is assisted for a second time by R. Daneel Olivaw. (The "R" stands for "Robot".) He's fully humanoid, and despite the Solarians' expertise with robots, able to conceal his robotic nature completely. Lije deals with uncooperative authorities, assassination attempts on him and his witnesses, and a world completely different from his own as he attempts to solve the mystery.

Tropes used in The Naked Sun include:

Androids and Detectives - Less emphasis on this than there was in The Caves of Steel, but it's still present.
Bluffing the Murderer - In the finale, as he lists the motives and means and even opportunities of everyone involved, without focusing on the murderer specifically.
Blue and Orange Morality: The whole morals of Solaria seem to be built around personal contact being obscene.
Designer Babies - All children on Solaria are grown in test tubes and vats, screened for genetic problems, raised collectively and taught to dislike human presense. This is considered a repugnant job, but Delmarre took it out of his strong sense of duty.
Electronic Speech Impediment - Robots who partially violate a Law can suffer damage to their speech programming; one robot who serves poisoned tea unknowingly ends up with a stammer. This is why there was a broken robot at the crime scene - its arm was the murder weapon, and when it realized what had happened, the First Law violation caused its programming to snap.
Fair Play Whodunnit
Found the Killer, Lost the Murderer - Inverted. Lije catches the murder but deliberately lets the killer go free.
Innocent Fanservice Girl: Since Solarians despise human contact they communicate with very realistic holograms (called "viewing", as opposed to "seeing"). When Baley first wants to talk with Gladia, she appears naked on the hologram; she doesn't understand his embarrassment, since it's just "viewing" -- it's not as though he's there in person or anything. Daneel tells her to put some clothes on.
It Is Pronounced "Tro-PAY" - Gladia's name is pronounced glah-DIE-ah.
Make It Look Like an Accident - All of the attempted murders on Lije and his assistants are carried out through robotic means, despite the laws of robotics! See "Three Laws"-Compliant below.
Nightmare Fuel - In-universe examples. Centuries of being cooped up inside walled cities have given all Earthmen intense agoraphobia (fear of the outdoors), to the point that Daneel fears Lije will panic just from seeing out an open window, and flatly refuses to let him open the top of the convertible car they're in when they first arrive on the planet. On the other side of the spectrum, there's Solarians. They've isolated themselves from others for so long that they have Anthropopobia (fear of other people). In one case, it's so bad that one of the characters mentally regress nearly to infancy at the thought of being in the same room as another person.
Perfect Poison: Subverted. The poisoner uses too much poison, and the victim vomits it up before it can kill him.
Please Put Some Clothes On: Gladia's first interview.
"Three Laws"-Compliant - Naturally. There are no mis-programmed robots in this book. However, there is an unspoken assumption built into the laws of robotics that becomes highly significant. Law One: "A robot may not knowingly injure a human being or, through inaction, knowingly allow a human being to come to harm." Basically, a robot can perform an innocuous task, individually harmless, which taken along with a number of other tasks performed by other robots or people, would cause a human being serious harm or death.
Title Drop
Torture Always Works - Subverted. Lije set up events so that the person he pins the murder on will think he's about to be in physical proximity with another human, an idea that revolts him even more than most other Solarians. This causes him to "confess" to his crimes, but he never actually confesses to the murder. In the end, it turns out he wasn't actually the murderer.
Zeerust - Dr. Delmarre and his assistant were still working on how to create children without two people having sex, thousands of years into the future.
- Maybe not Zeerust. This would involve not just artificial insemination, but also artificial wombs. After all, having another person inside you is the ultimate in proximity.