A theoretical and atomistic study of diffusion and stability of a pure element hollow nanosphere and nanotube is performed. The shrinkage via the vacancy mechanism of these hollow nano-objects is described analytically. Using Gibbs-Thomson boundary conditions an exact solution of the kinetic equation in quasi steady-state at the linear approximation is obtained. The collapse time as a function of the geometrical sizes of the hollow nano-objects is determined. Kinetic Monte Carlo simulation of the shrinkage of these nano-objects is performed: it confirms the predictions of the analytical analysis. Next, molecular dynamics simulation in combination with the embedded atom method is used to investigate diffusion by the vacancy mechanism in a Pd hollow nanosphere and nanotube. It is found that the diffusion coefficient in a Pd hollow nanosphere and nanotube is larger near the inner and external surfaces compared with the middle part of a nanoshell. The molecular dynamics results provide quite a strong but indirect argument that a real pure element hollow nanosphere and nanotube may not shrink as readily via the vacancy mechanism as compared with the predictions of the analytical analysis and kinetic Monte Carlo simulations.