Dual representations of convex bodies and their polars
The author proves the following Krein-Milman type theorem [see e.g. \textit{K. Leichtweiss}, ``Konvexe Mengen'', Berlin: Dt. Verlag der Wissenschaften (1980; Zbl 0427.52001)]: For a convex body \(C\) in a real Hausdorff locally convex space with \(0\in \text{ int }C\) the following representations are equivalent: \[ C= \bigcap_{x\in\Gamma(C)} H(x) \quad \text{and} \quad C^0 =\text{clco} \exp C^0, \] where \(H(x)\) denotes the supporting half-space in the tangency point \(x\in \Gamma (C)\), \(\exp C^0\) denotes the set of (weakly*) exposed points of the polar \(C^0\) of \(C\), and clco denotes the (weakly*) closed convex hull.
scientific article; zbMATH DE number 1023827