What mathematical ideas can be used to define the highest point on a planet (for example, the top of Mt Everest for Earth)? If we think of the planet as a solid, one idea is that the highest point is the point farthest from the centroid. However, this method would give strange answers for Earth because of the fact that the Earth looks more like an ellipsoid that is bigger in the east-west direction than in the north-south direction than it does like a sphere. Is the idea of a "sea level" important?
Conventionally the height about the geoid is used (this is more or less the same as height above sea level). Note that the geoid is within ±100m of the reference ellipsoid for the earth. Using distance from the center of the earth results in some mountain in the Andes being "higher" than Mt Everest. A more nonsensical result is that Mt McKinley is "lower" than Key West (by 7km)!