"in the world" usually means the complete set of things and people on planet earth. Everything. It is often used to emphasize something: "Antarctica is the coldest place in the world," or "The best doctor in the world," or "The only place in the world where you can get waffles and fried chicken served together." "around the world" can mean all the places in the world, referred to collectively, together, and can also mean that something has traveled almost everywhere on the planet. ("I've been around the world, and I can't find my baby" is a song by Lisa Stansfield. She is sad that she has looked everywhere for her lost love but did not find him or her. This example is more poetic and less literal, but it gives the flavor of this phrase well. "across the world" can also mean that something has traveled everywhere on the planet (like "around the world", but is most often used with the verb "spread". ("News of the pandemic spread across the world quickly, like the virus itself.")