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(which divide the space into the half-spaces) that contain . This yields an affine subspace . For each half-space where the hyperplane does not contain , we consider the intersection of the interior of those half-spaces. This yields an open set . Clearly, . Since is an extreme point of and is relatively open, it follows that must be 0-dimensional and . If was not 0-dimensional, would be the inner point of (at least) a line, which contradicts being an extreme point. Since every construction of chooses

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