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On the uniqueness of gravitational centre. (English) Zbl 1136.52001

Summary: The dual volume of order \(\alpha\) of a convex body \(A\) in \(\mathbb R^{n}\) is a function which assigns to every \(a \in A\) the mean value of \(\alpha\)-power of distances of a from the boundary of \(A\) with respect to all directions. We prove that this function is strictly convex for \(\alpha > n\) or \(\alpha <0\) and strictly concave for \(0< \alpha < n\) (for \(\alpha =0\) and for \(\alpha = n\) the function is constant). It implies that the dual volume of a convex body has the unique minimizer for \(\alpha > n\) or \(\alpha <0\) and has the unique maximizer for \(0< \alpha < n\). The gravitational centre of a convex body in \(\mathbb R^{3}\) coincides with the maximizer of dual volume of order \(2\), thus it is unique.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A40 Inequalities and extremum problems involving convexity in convex geometry
51P05 Classical or axiomatic geometry and physics
85A25 Radiative transfer in astronomy and astrophysics
86A20 Potentials, prospecting
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