Proof of L\'aszl\'o Fejes T\'oth's zone conjecture

A zone of width $\omega$ on the unit sphere is the set of points within spherical distance $\omega \mathrm{/}2$ of a given great circle. We show that the total width of any collection of zones covering the unit sphere is at least $\pi$, answering a question of Fejes T'oth from 1973.