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Shephard (Canad. J. Math. 26 (1974), 302-321) proved a decomposition theorem for zonotopes yielding a simple formula for their volume. In this note, we prove a generalization of this theorem yielding similar formulae for their intrinsic volumes. We use this result to investigate geometric extremum problems for zonotopes generated by a given number of segments. In particular, we solve isoperimetric problems for 𝑑-dimensional zonotopes generated by 𝑑 or 𝑑 + 1 segments, and give asymptotic estimates for the solutions of similar problems for zonotopes generated by sufficiently many segments. In addition, we present applications of our results to the 𝓁 1 polarization problem on the unit sphere and to a vectorvalued Maclaurin inequality conjectured by Brazitikos and McIntyre in 2021.

We shall give the maximum radius of fourteen congruent, nonoverlapping spheres in a cube.

Hadarics et. al. gave a Mathematical Model for Distributed Vulnerability Assessment. In this model the extent of vulnerability of a specific company IT infrastructure is measured by the probability of at least one successful malware attack when the users behaviour is also incorporated into the model. The different attacks are taken as independent random experiments and the probability is calculated accordingly. The model uses some input probabilities related to the characteristics of the different threats, protections and user behaviours which are estimated by the corresponding relative frequencies. In this paper this model is further detailed, improved and a numerical example is also presented.