Abstract
The mean volume of boxes circumscribed about a convex bodyK of given volume is a minimum whenK is a ball. This follows from a more general inequality, where the volume of circumscribed boxes is replaced by the product of quermassintegrals of the projections ofK on appropriate lower dimensional subspaces.
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Chakerian, G.D. The mean volume of boxes and cylinders circumscribed about a convex body. Israel J. Math. 12, 249–256 (1972). https://doi.org/10.1007/BF02790751
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DOI: https://doi.org/10.1007/BF02790751