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In recent years, the advance of novel chemical growth techniques has led to the fabrication of complex, three-dimensional magnetic nanostructures. The corners and edges of such realistic geometries are generally not sharp but rounded. In a previous article we have argued that high demagnetization fields in the vicinity of sharp edges lead to the formation of an asymmetric vortex state in pyramidal-shaped magnetic shell structures. The asymmetric vortex state is potentially interesting with respect to future magnetic memory devices. In this work a micromagnetic model is used to investigate the effect of rounded corners and edges on the magnetic reversal process within these pyramidal-shaped magnetic shell structures. In particular, we explore the degree of rounding, which has to be introduced in order to suppress the asymmetric vortex state. Another emphasis is placed on the magnetic reversal of (quasi-)homogeneous states within these structures. We demonstrate that the rounding of corners significantly reduces the coercivity. This complies with former studies on cuboidal structures, which suggest the important effect of corners on the magnetic reversal of homogeneous magnetic states. The present study uses a finite-element discretization for the numerical solution of the micromagnetic equations, which provides flexibility with respect to the modeling of complex shapes. In particular, this method is very accurate with respect to structures with a smooth surface.