A more complete version of a minimax theorem

In this paper, we present a more complete version of the minimax theorem established in [7]. As a consequence, we get, for instance, the following result: Let $X$ be a compact, not singleton subset of a normed space $(E,\|\cdot\|)$ and let $Y$ be a convex subset of $E$ such that $X\subseteq \overline {Y}$. Then, for every convex set $S\subseteq Y$ dense in $Y$, for every upper semicontinuous bounded function $\gamma:X\to {\bf R}$ and for every $\lambda>{{4\sup_X|\gamma|}\over {diam(X)}}$, there exists $y^*\in S$ such that the function $x\to \gamma(x)+\lambda\|x-y^*\|$ has at least two global maxima in $X$.