The extrema of \(q\)- and dual \(q\)-quermassintegrals for the asymmetric \(L_p\)-difference bodies (Q6634342)

Let \(K\) be a convex body in \({\mathbb R}^n\), containing the origin in the interior, and let \(h(K,\cdot)\) its support function. For \(p\ge 1\) and \(\tau\in[-1,1]\), the asymmetric \(L_p\)-difference body \(\Delta_p^\tau K\) of \(K\) is defined by \N\[\N h^p(\Delta_p^\tau K,\cdot) = f_1(\tau)h^p(K,\cdot) + f_2(\tau)h^p(-K,\cdot),\N\]\N\[\Nf_1(\tau)= \frac{(1+\tau)^p}{(1+\tau)^p+(1-\tau)^p}, \quad f_2(\tau)=f_1(-\tau).\N\]\N\textit{W. Wang} and \textit{T. Ma} [Proc. Am. Math. Soc. 142, No. 7, 2517--2527 (2014; Zbl 1295.52008)] have introduced these bodies and have proved volume inequalities for them. These inequalities are here extended to \(q\)-quermassintegrals and dual \(q\)-quermassintegrals (without \(q\), this was done before). The proofs make use (among other results established earlier) of Brunn-Minkowski type inequalities for the \(q\)-quermassintegrals and dual \(q\)-quermassintegrals.