  The Brownian force model (BFM) is the mean-field model for the avalanches of
an elastic interface slowly driven in a random medium. It describes the
spatio-temporal statistics of the velocity field, and, to some extent is
analytically tractable. We extend our previous studies to obtain several
observables for the BFM with short range elasticity, related to the local jump
sizes $S(x)$ and to the avalanche spatial extension in $d=1$, or the avalanche
span in $d>1$. In $d=1$ we consider both driving (i) by an imposed force (ii)
by an imposed displacement "at the tip" and obtain in each case the mean
spatial shape $\langle S(x) \rangle$ at fixed extension, or at fixed seed to
edge distance. We find that near an edge $x_e$, $S(x) \simeq \sigma |x-x_e|^3$
where $\sigma$ has a universal distribution that we obtain. We also obtain the
spatiotemporal shape near the edge. In $d>1$ we obtain (i) the mean shape
$\langle S(x_1,x_\perp) \rangle$ for a fixed span, which exhibits a non-trivial
dependence in the transverse distance to the seed $x_\perp$ (ii) the mean shape
around a point which has not moved, $\langle S(x) \rangle_{S(0)=0}$, which
vanishes at the center as $|x|^{b_d}$ with non trivial exponents, $b_1=4$,
$b_2=2 \sqrt{2}$ and $b_3=\frac{1}{2} (\sqrt{17}-1)$. We obtain the probability
distributions in any $d$ for the maximal radius of an avalanche and the minimal
distance of approach to a given point, as well as the probability of not
hitting a cone in $d=2$. These results equivalently apply to the continuum
limit of some branching diffusions, as detailed in a companion paper.
