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By Pickert G.

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If C is convex then rel int C and C are convex. 6 (13) Superfluous mingling of terms as in relatively nonempty set would be an unfortunate consequence. From the opposite perspective, some authors use the term full or full-dimensional to describe a set having nonempty interior. 3), although relative closure is superfluous. 1] 40 CHAPTER 2. CONVEX GEOMETRY (a) R2 (b) (c) Figure 12: (a) Closed convex set. (b) Neither open, closed, or convex. 3). 49] are convex. (c) Open convex set. 3] int{x} = ∅ = ∅ the empty set is both open and closed.

14 54 CHAPTER 2. CONVEX GEOMETRY PT 3 R3 R B P T (B) x PTx Figure 19: Linear noninjective mapping P T x = A†Ax : R3 → R3 of three-dimensional Euclidean body B has affine dimension 2 under projection on rowspace of fat full-rank matrix A ∈ R2×3 . Set of coefficients of orthogonal projection T B = {Ax | x ∈ B} is isomorphic with projection P (T B) [sic]. , T represented by skinny-or-square full-rank matrices. (Figure 18) An important consequence of this fact is: Affine dimension, of any n-dimensional Euclidean body in domain of operator T , is invariant to linear injective transformation.

2. VECTORIZED-MATRIX INNER PRODUCT R2 (a) 51 R3 (b) Figure 17: (a) Cube in R3 projected on paper-plane R2 . Subspace projection operator is not an isomorphism because new adjacencies are introduced. (b) Tesseract is a projection of hypercube in R4 on R3 . More generally, vwT in (38) may be replaced with any particular matrix Z ∈ Rp×k while convexity of set Z , C ⊆ R persists. Further, by replacing v and w with any particular respective matrices U and W of dimension compatible with all elements of convex set C , then set U TCW is convex by the inverse image theorem because it is a linear mapping of C .

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