# Commutativity of inf and sup

Under which conditions on the potential $\Pi(u,p)$ does the following hold?

$$\inf_{u} \sup_{p} \Pi(u,p) = \sup_{p}\inf_{u} \Pi(u,p)$$

It is obvious that both expressions have the same stationary points and that if there exists only one stationary point, that point is the same, but is this commutation always valid (under the assumption that the first inf-sup is valid)?

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Do you mean $=$ instead of $\iff$? –  martini Nov 15 '12 at 14:54
Yes, thats right. Thanks! –  Carl Nov 15 '12 at 14:58
This will work very seldom. Even if the $\sup$ and $\inf$ are taken over sets of size 2, we can have $\ne$. Cosider $\Pi(u,p) = \delta_{up}$, $u,p \in \{0,1\}$. –  martini Nov 15 '12 at 15:17
Most minimax results that I have seen have $\Pi$ convex in $u$ and concave in $p$ (or relaxations thereof) such as Theorem 2, ch. XVI, section 5 of Kantorovich & Akilov, "Functional analysis". Other names to look for are von Neumann, Ky Fan. The result is connected with fixed-point theorems. –  copper.hat Nov 15 '12 at 16:52
But what if $\Pi$ is concave in $u$ and convex in $p$ for instance? I should point out that I am by no means a mathematician, I am working working on a Stokes flow problem and need to commute the inf and sup on a potential arising from a minimization problem where $p$ is a Lagrange multiplier. I omitted that information in order to keep the post as general as possible. Any help is greatly appreciated. –  Carl Nov 15 '12 at 17:01