The Karenins continue to live together, but it's a sham marriage. Anna keeps on seeing Vronsky. No one likes the situation, but they all think it's temporary.
Karenin feels that Anna's thing for Vronsky will pass, and they will return to the life they once led.
Anna feels that it will all get better on its own. She doesn't know how the situation will improve, but she is sure that it'll be okay in the end.
Vronsky also thinks something's going to happen outside of himself that'll solve their issues.
In the middle of winter, in his capacity as a military officer, Vronsky escorts a foreign prince around all the sights of St. Petersburg. The prince is a great lover of pleasure, so one of Vronsky's duties is to introduce the guy to all the great nightlife Russia has to offer.
The prince is psyched, but Vronsky notices a change in himself. Where once, all he would have wanted to do would be to eat caviar and sleep with ballerinas, he now sees the Prince's activities as pointless and boring.
What's even worse is that Vronsky sees his own arrogance and snobby behavior in the Prince, and that disturbs him. And because he's the Prince's social inferior, the guest's arrogance is directed at Vronsky.
Vronsky experiences first hand what it's like to be treated the way Vronsky himself treats his social inferiors. Vronsky is both ashamed of himself and humiliated by the Prince's behavior towards him. He's glad to see the Prince leave.