Abstract
The dual volume of order α of a convex body A in Rn is a function which assigns to every a ∈ A the mean value of α-power of distances of a from the boundary of A with respect to all directions. We prove that this function is strictly convex for α > n or α < 0 and strictly concave for 0 < α < n (for α = 0 and for α = n the function is constant). It implies that the dual volume of a convex body has the unique minimizer for α > n or α < 0 and has the unique maximizer for 0 < α < n. The gravitational centre of a convex body in R3 coincides with the maximizer of dual volume of order 2, thus it is unique.
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Herburt, I. On the Uniqueness of Gravitational Centre. Math Phys Anal Geom 10, 251–259 (2007). https://doi.org/10.1007/s11040-007-9031-6
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DOI: https://doi.org/10.1007/s11040-007-9031-6