Refined convexity and special cases of the Blaschke-Santalo inequality (Q2770421)

The authors use the following modified version of convexity of functions: If \(\varphi\) is a real, continuous strictly monotonic function defined on an interval \(I\), then \(M:I^2 \rightarrow I\) defined as NEWLINE\[NEWLINE M(x,y):=\varphi^{-1}((\varphi(x) + \varphi(y))/2) NEWLINE\]NEWLINE is called a quasi-arithmetic mean. NEWLINENEWLINENEWLINESimilarly NEWLINE\[NEWLINE M^{(\lambda)}(x,y):= \varphi^{-1}(\lambda \varphi (x) + (1-\lambda)\varphi (y)) NEWLINE\]NEWLINE for \(\lambda \in [0,1]\) is the weighted version of \(M\). NEWLINENEWLINENEWLINEFor any two quasi-arithmetic means \(M\), \(N\), a function \(f:I \rightarrow J\) is called \((M,N)\)-convex if it satisfies NEWLINE\[NEWLINE f(M^{\lambda}(x,y)) \leq N^{\lambda}(f(x),f(y)) NEWLINE\]NEWLINE for all \(x,y \in I\) and for all \(\lambda \in [0,1]\). NEWLINENEWLINENEWLINEUsing this modified notion of convexity the authors derive the \(\|. \|_p\)-norm version of the classical Blaschke-Santalo inequality for polar volumes. They also give analogs for the \((p,q)\)-substitution norm.