Inequalities for dual quermassintegrals of mixed intersection bodies (Q1770021)

Continuing some of their recent work on inequalities for mixed intersection bodies, the authors prove Brunn-Minkowski and Alexandroff-Fenchel inequalities for dual quermassintegrals of such bodies. The results are dual to inequalities for projection bodies established by Lutwak. Let \(I(K_1,\ldots,K_{n-1})\) denote the mixed intersection body of the star bodies \(K_1,\ldots,K_{n-1}\) and let \(\widetilde{W}_i\) be the \(i\)th dual quermassintegral. Let further \(\widetilde{+}\) denote radial addition of star bodies. Inequalities established are \[ \begin{aligned} \widetilde{W}_i ( I(K_1,\ldots,K_{n-1}) ) &\leq \prod_{j=1}^r \widetilde{W}_i ( I(K_j,\ldots,K_j,K_{r+1},\ldots,K_{n-1}) )\\ \text{ for \(0\leq i<n\), \(0<j<n-1\) and \(0<r\leq n-1\) and} \widetilde{W}_i ( I( K \widetilde{+} L ))^{1/(n-i)(n-1)} &\leq \widetilde{W}_i ( I( \alpha K \widetilde{+} (1-\alpha) L ))^{1/(n-i)(n-1)}\\ &+ \widetilde{W}_i ( I( (1-\alpha) K \widetilde{+} \alpha L ))^{1/(n-i)(n-1)} \end{aligned} \] for \(0\leq i <n\) and \(0\leq \alpha\leq 1\).