In the Sion's minimax thorem we assume that $\ X$ is a compact convex subset of a linear topological space and $\ Y$ is a convex subset of a linear topological space. In addition we say that $\ f$ is a real-valued function s.t.

$\ f(x,\cdot)$ upper semicontinuous and quasiconvex function on $\ Y$, $\forall x\in X$, and

$\ f(\cdot,y)$ lower semicontinuous and quasi-convex on $\ X$, $\forall y\in Y$

Then $\min_{x\in X}\sup_{y\in Y} f(x,y)=\sup_{y\in Y}\min_{x\in X}f(x,y).$

I can understand why compcatness is needed (we can choose X and Y to be the real plane and take any increase function as f)

But I don't see why convexity and semi-continuity are neccessary?

  • $\begingroup$ Are you familiar with any proofs of the result? $\endgroup$ – Pete Caradonna Mar 8 '18 at 0:45

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