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It is found that for objects possessing small surface structures with differing radii of curvature the secondary electron emission (SEE) yield may be significantly higher than for objects with smooth surfaces of the same material. The effect is highly pronounced for surface structures of nanometer scale, often providing a more than 100% increase of the SEE yield. The results also show that the SEE yield from surfaces with structure does not show a universal dependence on the energy of the primary, incident electrons as it is found for flat surfaces in experiments. We derive conditions for the applicability of the conventional formulation of SEE using the simplifying assumption of universal dependence. Our analysis provides a basis for studying low-energy electron emission from nanometer structured surfaces under a penetrating electron beam important in many technological applications.