I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the dual Banach space $X^*$. I have a feeling that such points already have some standard names (I would suggest corner points, conical points or something like that) but cannot find a corresponding definition in the literature. Please help!
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asked Mar 17, 2021 at 8:13
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1$\begingroup$ I don't know if it's a standard name, but to me it seems natural to call such a point a "vertex". $\endgroup$– Yoav KallusCommented Mar 17, 2021 at 12:58
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$\begingroup$ @YoavKallus Good suggestion especially in terms of duality between cells of maximal and minimal dimension. Thank you. $\endgroup$– Taras BanakhCommented Mar 17, 2021 at 13:37
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1$\begingroup$ In "On the Geometry of Numerical Ranges" by Radjabalipour and Radjavi, Pacific J. Math, Vol. 61, No. 2, 1975 where they worked in the two dimensional case, they called such points sharp points; unfortunately in the applied math literature "sharp points" seems have to be taken to have a different meaning in higher dimensions. $\endgroup$– Willie WongCommented Mar 17, 2021 at 16:04
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1$\begingroup$ In "Sets of matrices with given joint numerical range" by N. Krupnik, I.M. Spitkovsky / Linear Algebra and its Applications 419 (2006) 569–585, you have at least a precedent for calling it conical point. The definition they gave is that given a closed set $F$, $x\in F$ is a conical point if and only if there exists a proper, closed, convex cone $K$ such that $F\subset x + K$. $\endgroup$– Willie WongCommented Mar 17, 2021 at 16:08
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$\begingroup$ @WillieWong What is a proper cone? As I understand, a hyperplane is not a proper cone? $\endgroup$– Taras BanakhCommented Mar 17, 2021 at 17:59
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