Consider a set $K\subseteq \mathbb{R}^d$ (not necessarily closed and/or convex). Let $k:=(k_1,...,k_i,...,k_d)$ be a generic element of $K$.

Consider the function $h_K:\mathbb{R}^d\rightarrow \mathbb{R}$ $$ h_K(u):=\sup_{k\in K} \sum_{i=1}^d k_iu_i $$ with $u:=(u_1,...,u_i,...,u_d)\in \mathbb{R}^d$.

Under which sufficient conditions on $K$ $\sup_{k\in K} \sum_{i=1}^d k_iu_i=\max_{k\in K} \sum_{i=1}^d k_iu_i$?

  • $\begingroup$ Do you think there is a simple answer on this question? $\endgroup$ – gerw Apr 22 '16 at 17:15
  • $\begingroup$ I have no idea. $\endgroup$ – STF Apr 22 '16 at 17:33
  • $\begingroup$ Just one set of sufficient conditions would be enough. $\endgroup$ – STF Apr 22 '16 at 18:11
  • 1
    $\begingroup$ One sufficient condition is that $K$ is compact. $\endgroup$ – gerw Apr 22 '16 at 18:49

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