  We prove a positive volume theorem for asymptotically AdS spacetimes: the
maximal volume slice has nonnegative vacuum-subtracted volume, and the
vacuum-subtracted volume vanishes if and only if the spacetime is identically
pure AdS. Under the Complexity=Volume proposal, this constitutes a positive
holographic complexity theorem. The result features a number of parallels with
the positive energy theorem, including the assumption of an energy condition
that excludes false vacuum decay (the AdS weak energy condition). Our proof is
rigorously established in broad generality in four bulk dimensions, and we
provide strong evidence in favor of a generalization to arbitrary dimensions.
Our techniques also yield a holographic proof of Lloyd's bound for a class of
bulk spacetimes. We further establish a partial rigidity result for wormholes:
wormholes with a given throat size are more complex than AdS-Schwarzschild with
the same throat size.
