  A cornerstone in convex analysis is the crucial relationship between
functions and their convex conjugate via the Fenchel-Young inequality. In this
dual variable setting, the maximal monotonicity of the contact set $
\big\{(x,y) \ \big| \ f(x) + f^*(y) = \langle x,y \rangle \big\}$ is due to the
involution $f^{**} = f$ holding for convex lower-semicontinuous functions
defined on any Hilbert space.
  We investigate the validity of the cyclic version of involution and maximal
monotonicity for multiple (more than two) convex functions. As a result, we
show that when the underlying space is the real line, cyclical involutivity and
maximal monotonicity induced by multi-conjugate convex functions continue to
hold as for the dual variable case. On the other hand, when the underlying
space is multidimensional, we show that the corresponding properties do not
hold in general unless a further regularity assumption is imposed. We provide
detailed examples that illustrate the significant differences between dual- and
multi-conjugate convex functions, as well as between uni- and multi-dimensional
underlying spaces.
