Star Lord

Star Lord has landed on a deserted planet with one space policeman. He is moving around the planet, painting a line along his path, claiming any land which is surrounded by paint (the part containing less unclaimed area). The policeman is trying to restrict the total land claimed by Star Lord as much as possible. If he encounters him, Star Lord gets arrested and can not continue painting anymore. Can you prove that the policeman has a strategy, which prevents Star Lord from claiming more than 50% of the planet's surface? If there are two policemen on the planet, can you prove that they have a strategy, which prevents Star Lord from claiming more than 25% of the surface?

Remark: We assume that Star Lord and the policemen are moving with the same speed, take decisions in real time and are fully aware of everybody's locations. Their initial positions are arbitrary and the planet is a perfect sphere.


SOLUTION

The strategy for one policeman is the following: in the beginning take a grand circle on the sphere such that Star Lord and the policeman are symmetric with respect to it. Now if the policeman mirrors Star Lord's movement with respect to the grand circle, he will never lit him leave his hemi-sphere.
The strategy for two policemen is the following: in the beginning choose an axis for the planet, so that Star Lord and the two policemen lie on the same parallel. Now the policemen just have to keep up the same latitude as Star Lord and move towards him with respect to their longitudes, squeezing him inbetween. If Star Lord travels in total X degrees with respect to the planet's latitude, then the two policemen combined would have traveled 2X degrees with respect to the planet's latitude. However, if they do not catch him, then 2X will be smaller than 360 (the total degrees on the latitude) and therefore X < 180. This implies that the area encompassed by Star Lord can't stretch between two meridians of difference larger than 90 degrees (in order to be encompassed, Star Lord must travel around it in both West and East directions). Therefore it will be contained in a quarter slice of the planet.