Inequalities for generalized intersection bodies (Q536494)

In [Trans. Am. Math. Soc. 345, 777--801 (1994; Zbl 0812.52005)] \textit{G. Zhang} defined the \textit{intersection body of order} \(i\in\{1,\dots,n-1\}\) of a star body \(L\) as the centered star body \(I_iL\) whose radial function is given by \[ \rho_{I_iL}(u)=\frac{1}{n-1}\int_{S^{n-1}\cap u^{\bot}}\rho_L(v)^idv,\quad u\in S^{n-1}. \] Denoting by \(\widetilde{W}_i(K)\) the dual quermassintegral of \(K\), the authors show that if \(K\) is an intersection body (respectively, a centered star body) and \(L\) is a star body in \({\mathbb R}^n\) such that \(I_iK\subset I_iL\) (respectively, \(I_iK=I_iL\)), then \(\widetilde{W}_{n-i-1}(K)\leq\widetilde{W}_{n-i-1}(L)\) for all \(1\leq i\leq n-1\), with equality if and only if \(K=L\). These two results generalize, respectively, a Busemann-Petty type theorem by Lutwak (1988) and an extension of the Funk section theorem by Lv and Leng (2007): for \(K\) and \(L\) as before, if the volume of any (\(n-1\))-dimensional central section of \(K\) is smaller than the volume of the corresponding central section of \(L\) (respectively, if \(IK=IL\)), then \(\text{vol}(K)\leq\text{vol}(L)\), equality attained if and only if \(K=L\).