  Quantifying the universality of avalanche observables beyond critical
exponents is of current great interest in theory and experiments. Here, we
improve the characterization of the spatio-temporal process inside avalanches
in the universality class of the depinning of elastic interfaces in random
media. Surprisingly, at variance with the temporal shape, the spatial shape of
avalanches has not yet been predicted. In part this is due to a lack of an
analytically tractable definition: how should the shapes be centered? Here we
introduce such a definition, accessible in experiments, and study the mean
spatial shape of avalanches at fixed size centered around their starting point
(seed). We calculate the associated universal scaling functions, both in a
mean-field model and beyond. Notably, they are predicted to exhibit a cusp
singularity near the seed. The results are in good agreement with a numerical
simulation of an elastic line.
