Sion's minimax theorem assumes that $f:X\times Y\to\mathbb{R}$ is being minimized w.r.t. $x$ and maximized w.r.t. $y$, where at least one of $X,Y$ is compact (additional (quasi)convexity and semi-continuity properties are assumed).
Question: under the assumption that $f$ is linear and continuous in each argument and $X,Y$ are both convex, closed and bounded (but not compact), does the conclusion of Sion's theorem still hold?
