TITLE: Is there an intuition for cyclic monotonicity?
QUESTION [5 upvotes]: Cyclic monotonicity says that if we have a correspondence, $x(w)$, the $x$ is cyclically monotone in $w$ if for a finite sequence $w_1,\cdots w_k$ and $x^*(w_i)\in x(w_i)$, we have $\sum_{i=1}^k (w_i-w_{i+1})\cdot x^*(w_i) \leq 0$ (I guess this is technically the definition for cyclically monotone increasing).

I am wondering if there is some intuition (or clear) explanation of what this says. 

To me, it seems to say that when $w_{i+1} >w_i$ then $x_i > x_{i+1}$, in some sort of "general" sense.

REPLY [3 votes]: R.T. Rockefellar was the convex analyst who showed that a (multivalued) linear operator is the subdifferential of a convex function iff the operator is cyclically monotone.  To quote one of his papers on that topic:

The cyclic  monotonicity  condition  can  be  viewed   heuristically   as  a discrete  substitute  for   two  classical   conditions:  that  a   smooth  convex function   has   a   positive   semi-definite   second   differential,   and  that  all circuit  integrals   of  an  integrable  vector  field  must  vanish.