On mixed brightness-integrals (Q2855959)

\textit{N. Li} and \textit{B. Zhu} recently introduced in [J. Korean Math. Soc. 47, No. 5, 935--945 (2010; Zbl 1201.52007)] the mixed brightness-integrals \(C(K_1,\ldots, K_n)\) and \(C_s(K_1,\ldots, K_n)\) (\(s\neq 0\)) for any convex bodies \(K_i\) (\(i=1,2,\ldots, n\)) in an Euclidean \(n\)-dimensional space.NEWLINENEWLINEOne of the main results of the paper under review asserts that if \(K_i\) (\(i=1,2,\ldots, n\)) are convex bodies with centroid at the origin of \(\mathbb R^n\), \(0<p<1\) and \(1/p+1/q=1\), then NEWLINE\[NEWLINE C(K_1,\ldots, K_n)^{1/p}C(K_1^*,\ldots, K_n^*)^{1/q} \leq 2^{-n} \omega _n \prod _{j=1}^n R_j^{1/q} (R_j^*)^{1/p}, NEWLINE\]NEWLINE with equality if and only if all \(K_j\) are \(n\)-balls centered at the origin. Here, \(\omega_n\) denotes the volume of the unit ball in \(\mathbb R^n\), \(R_j\) and \(R_j^*\) are the out-radius of the mixed projection body \(\prod K_j\) and of its polar dual \(\prod ^* K_j\), respectively.NEWLINENEWLINEA similar statement is valid for the mixed brightness-integrals of order \(s\neq 0\).