Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms (Q2053448)

On the set \({\mathcal S}^n_o\) of star bodies in \({\mathbb R}^n\), one has the radial addition \(\widetilde +\), the radial Blaschke addition \(\widetilde\#\), and the star duality, all defined via simple operations with the radial function. A map \(\Psi:{\mathcal S}^n_o\to{\mathcal S}^n_o\) is a radial Blaschke--Minkowski homomorphism if it is continuous with respect to the radial metric, \(SO(n)\) equivariant, and satisfies \(\Psi(K\widetilde\# L)= \Psi(K)\widetilde+\Psi(L)\) for all \(K,L\in {\mathcal S}^n_o\). Starting from \(\Psi\), one can define the \(j\)th mixed radial Blaschke--Minkowski homomorphism \(\Psi_j\). Let \(\Psi_j^o(K)\) be the star dual of \(\Psi_j(K)\), and let \(\widetilde W_i\) denote the \(i\)th dual quermassintegral. The present paper states various inequalities for \(\widetilde W_i\circ\Psi_j^o\). Reviewer's remark. The proof of Lemma 3.1, and thus of Theorems 1.1 and 1.2, is based on the relation \[ \rho(M,u)^{n-i-1}\left[\rho(N,u)^{n-j-1}*\mu\right] =\left[\rho(M,u)^{n-i-1}*\mu\right]\rho(N,u)^{n-j-1},\] which is not correct (example: \(N\) a ball with center at the origin, \(M\) arbitrary).