Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton

By William Fulton

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CONVEX GEOMETRY Now let’s move to an ambient space of three dimensions. Figure 11(c) shows a polygon rotated into three dimensions. For a line to pass through its zero-dimensional boundary (one of its vertices) tangentially, it must exist in at least the two dimensions of the polygon. But for a line to pass tangentially through a single arbitrarily chosen point in the relative interior of a one-dimensional face on the boundary as illustrated, it must exist in at least three dimensions. Figure 11(d) illustrates a solid circular pyramid (upside-down) whose one-dimensional faces are line-segments emanating from its pointed end (its vertex ).

4 All orthants are self-dual simplicial cones. 1) Two affine sets are said to be parallel when one is a translation of the other. 1. 6 empty set versus empty interior Emptiness ∅ of a set is handled differently than interior in the classical literature. , paper in the real world. 5 An ordinary flat sheet of paper is an example of a nonempty convex set in R3 having empty interior but relatively nonempty interior. 1 relative interior We distinguish interior from relative interior throughout. 24] and it is always possible to pass to a smaller ambient Euclidean space where a nonempty set acquires an interior.

Figure 12. Given P , the generating list {xℓ } is not unique. Given some arbitrary set C ⊆ Rn , its convex hull conv C is equivalent to the smallest closed convex set containing it. 1 Example. Hull of outer product. 1) conv U U T | U ∈ RN ×k , U T U = I = A ∈ SN | I 0 , I , A = k ⊂ SN + (79) A This important convex body we call Fantope (after mathematician Ky Fan). 1) conv U U T | U ∈ RN , U T U = I = A ∈ SN | A 0, I , A =1 (80) In case k = N , the Fantope is identity matrix I . 1) of its convex hull.

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