Holographic Complexity from the Crofton's Formula in Lorentzian AdS${}_3$

We study the Crofton’s formula in the Lorentzian AdS${}_3$ and find that the area of a generic space-like two dimensional surface is given by the flux of space-like geodesics. The “complexity=volume” conjecture then implies a new holographic representation of the complexity in terms of the number of geodesics. Finally, we explore the possible explanation of this result from the standpoint of information theory.