Inequalities involving integrals of polar-conjugate concave functions (Q1386386)

An inequality of K. Mahler providing a lower bound for the product of the area of a plane convex figure and the area of its polar convex figure can be written in the form of integrals of polar-conjugate concave functions [see \textit{M. Meyer}, Monatsh. Math. 112, No. 4, 297-301 (1991; Zbl 0737.52002)]. The authors generalise this inequality for integrals of powers of polar-conjugate functions and use it to derive further inequalities for the product of volumes of a convex body \(K\) and the polar to \(K-z\) where \(z\) is the Santaló point of \(K\).