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By Boyer Ch. P.

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3]) through a single point relatively interior to a three-dimensional face on ∂ C . Let’s understand why by inductive reasoning. Figure 11(a) shows a vertical line-segment whose boundary comprises its two endpoints. For a line to pass through the boundary tangentially (intersecting no point relatively interior to the line-segment), it must exist in an ambient space of at least two dimensions. Otherwise, the line is confined to the same one-dimensional space as the line-segment and must pass along the segment to reach the end points.

12 Because of this Euclidean structure, all the known results from convex analysis in Euclidean space Rn carry over directly to the space of real matrices Rp×k . 5) is isometrically isomorphic with its vectorized range vec R(A) but not with R(vec A). 48 CHAPTER 2. 2 Definition. Isometric isomorphism. An isometric isomorphism of a vector space having a metric defined on it is a linear bijective mapping T that preserves distance; id est, for all x, y ∈ dom T Tx − Ty = x − y (39) Then the isometric isomorphism T is a bijective isometry.

Tradition [148] [232] recognizes only positive normal polarity in support function σY as in (108); id est, normal a , figure (a). But both interpretations of supporting hyperplane are useful. 70 CHAPTER 2. 17 at each point on its boundary. 1 Definition. Supporting hyperplane ∂H . The partial boundary ∂H of a closed halfspace that contains arbitrary set Y is called a supporting hyperplane ∂H to Y when the hyperplane contains at least one point of Y . [232, 11] Specifically, given normal a = 0 (belonging to H+ by definition), the supporting hyperplane to Y at yp ∈ ∂Y [sic] is ∂H− = = y | aT(y − yp ) = 0 , yp ∈ Y , aT(z − yp ) ≤ 0 ∀ z ∈ Y y | aTy = sup{aTz | z ∈ Y} (108) where normal a and set Y reside in opposite halfspaces.

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