Convex, Affine, Conic: Hulls | Convex, Affine, Conic Hulls |
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The hull of a boat rests in the water; it contains the occupants and cargo. Hulls are a fundamental concept in convex geometry. The convex hull of two points, for example, comprises those points and the line segment between them. Their affine hull is the unique line containing them, while the conic hull is the union of all rays, emanating from the origin, intersecting the convex hull of those two points. Any closed convex set containing no lines can be expressed as the convex hull of its extreme points and extreme rays. The conic hull of any finite-length list forms a polyhedral cone, the smallest closed convex cone that contains the list. Dimension of an arbitrary set S is the dimension of its affine hull, which is the same as dimension of the subspace parallel to that affine set. The subspace of symmetric matrices is the affine hull of the cone of positive semidefinite matrices. |
