Let $E$ be an infinite-dimensional real normed space and let $K\subset E$ be a weakly compact set such that, for each $\varphi\in E^*\setminus \{0\}$, there exists a unique $\tilde x\in K$ such that $$\varphi(\tilde x)=\inf_{x\in K}\varphi(x)\leq 0\ .$$ Question: does $K$ necessarily contain $0$ ?
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$\begingroup$ What about a circled centered at the origin in $\mathbb{R}^2$ ? $\endgroup$– an_ordinary_mathematicianCommented Oct 18 at 13:53
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$\begingroup$ @an_ordinary_mathematician : The space must be infinite dimensional. $\endgroup$– Iosif PinelisCommented Oct 18 at 13:54
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$\begingroup$ @IosifPinelis, Ok, I see in the embedding of the circle in an infinite dimensional space one might loose uniqueness of the minimizer. $\endgroup$– an_ordinary_mathematicianCommented Oct 18 at 13:56
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