[en] Single, double and triple ionization cross-sections of C_n⁺ clusters in collisions with helium atoms at intermediate velocity (2.6 a.u.) have been measured (n=1→5). The relative multi-ionization cross-sections (double over single and triple over single) are first increasing with n, then decreasing when going from n=4 to n=5. We show, on the basis of an independent atom and electron collisional treatment, that this effect is attributable to a change in the cluster shape. The role of the cluster compactness on relative multi-ionization cross-sections is pointed out here for the first time. (orig.)