Sion's minimax theorem is usually stated with a condition that one of the sets is compact, e.g. from Wikipedia,
Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex subset of a linear topological space. If $f$ is a real-valued function on $X \times Y$, with $f(x, \cdot)$ upper semicontinuous and quasi-concave on $Y$ for all $x \in X$, and $f( \cdot, y)$ lower semicontinuous and quasi-convex on $X$ for all $y \in Y$, then $$\inf_{x \in X}\sup_{y \in Y}f(x,y) = \sup_{y \in Y}\inf_{x \in X}f(x,y).$$
However, if we restrict to the case of finite-dimensional Euclidean spaces (and if necessary, strengthen the conditions to continuity and actual concavity/convexity rather than quasi-concavity/convexity), is it strictly necessary to require compactness rather than just boundedness? The geometric proofs of minimax theorems suggest that this should be the case, but I may be missing a subtle counterexample.
(I am considering using this result in the context of strong duality, so it would also suffice for me if it can be shown that the Clark-Duffin condition for strong duality https://www.pnas.org/doi/10.1073/pnas.75.4.1624 also holds without the requirement that the objective and constraint functions are continuous on closed convex sets.)