The Legendre ellipsoid \(\Gamma_2K\) of a star body \(K\) in Euclidean space \(\mathbb{R}^n\) can be defined by its support function \[h_{\Gamma_2 K}(u):= \left({n+2 \over V(K)}\int_K (u\cdot x)^2 dx\right)^{1\over 2}\] for \(u\in S^{n-1}\). The authors propose a new ellipsoid, \(\Gamma_{-2}K\), which can be defined by its radial function \[\rho_{\Gamma_{-2} K}(u):= \left({1\over V(K)} \int_{S^{n-1}} (u\cdot v)^2 h_K^{-1} (v)dS_K(v) \right)^{-{1\over 2}}\] for \(u\in S^{n-1}\); here \(S_K\) is the classical surface area measure of \(K\). The motivation comes from Lutwak’s dual Brunn-Minkowski theory: it is remarked that the Legendre ellipsoid \(\Gamma_2K\) properly belongs to that theory, while \(\Gamma_{-2}K\) is its dual counterpart in the classical Brunn-Minkowski theory.
The authors present a neat parallel development of the known inequalities for the Legendre ellipsoid \(\Gamma_2K\) (with new proofs) and their newly established counterparts for the ellipsoid \(\Gamma_{-2}K\). Among the interesting series of volume inequalities for the new ellipsoid there is, surprisingly, even one for which the dual classical counterpart is still an open problem.

Reviewer: Rolf Schneider (Freiburg i.Br.)

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