Traveled lengths statistic is a key quantity for characterizing stochastic
processes in bounded domains. For straight lines and diffusive random walks,
the average length of the trajectories through the domain is independent of the
random walk characteristics and depends only on the ratio of the volume domain
over its surface, a behavior that has been recently observed experimentally for
exponential jump processes. In this article, relying solely on geometrical
considerations, we extend this remarkable property to all d-dimensional random
curves of arbitrary lengths (finite or infinite), thus including all kind of
random walks as well as fibers processes. Integral geometry will be central to
establishing this universal property of random trajectories in bounded domains.
