  The differential cross section of exclusive diffractive vector meson
production in electron proton collisions carries important information on the
geometric structure of the proton. More specifically, the coherent cross
section as a function of the transferred transverse momentum is sensitive to
the size of the proton, while the incoherent, or proton dissociative cross
section is sensitive to fluctuations of the gluon distribution in coordinate
space. We show that at high energies the experimentally measured coherent and
incoherent cross sections for the production of $J/\Psi$ mesons are very well
reproduced within the color glass condensate framework when strong geometric
fluctuations of the gluon distribution in the proton are included. For $\rho$
meson production we also find reasonable agreement. We study in detail the
dependence of our results on various model parameters, including the average
proton shape, analyze the effect of saturation scale and color charge
fluctuations and constrain the degree of geometric fluctuations.
