Isoperimetric and Brunn-Minkowski inequalities for the \((p, q)\)-mixed geominimal surface areas (Q2135054)

For \(p,q\in{\mathbb R}\), a convex body \(K\in{\mathcal K}_o^n\) (the set of convex bodies in \({\mathbb R}^n\) containing \(o\) in the interior) and a star body \(L\subset{\mathbb R}^n\), the \((p,q)\)-mixed geominimal surface area is defined by \[ \widetilde G_{p,q}(K,L):=\omega_n^{-p/n}\inf\{n\widetilde V_{p,q}(K,Q,L)V(Q^*)^{p/n}: Q\in{\mathcal K}_o^n\}.\] Here \(\omega_n\) is the volume of the unit ball in \({\mathbb R}^n\), \(\widetilde V_{p,q}(K,Q,L)\) is the \(L_p\) dual mixed volume as introduced by \textit{E. Lutwak} et al. [Adv. Math. 329, 85--132 (2018; Zbl 1388.52003)], here called the \((p,q)\)-dual mixed volume, and \(Q^*\) is the polar body of \(Q\). For \(L=K\) and \(p=1\), one obtains Petty's geominimal surface area. The authors prove several sharp inequalities involving this \((p,q)\)-mixed geominimal surface area and the volume of \(K\) or \(L\), or both. One of the inequalites uses the \(L_p\) harmonic Blaschke combination.