  It is well-known that if a real valued function acting on a convex set
satisfies the $n$-variable Jensen inequality, for some natural number $n\geq
2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-variable Jensen
inequality as well. In other words, the arithmetic mean and the Jensen
inequality (as a convexity property) are both reducible. Motivated by this
phenomenon, we investigate this property concerning more general means and
convexity notions. We introduce a wide class of means which generalize the
well-known means for arbitrary linear spaces and enjoy a so-called reducibility
property. Finally, we give a sufficient condition for the reducibility of the
$(M,N)$-convexity property of functions and also for H\"older--Minkowski type
inequalities.
