v2007.09.17 - Convex Optimization

40 CHAPTER 2. CONVEX GEOMETRYwhere Q∈ R 3×3 is an orthogonal matrix, then the projection on S 3 + in R 6 is⎡PX = Q⎣λ 1 0λ 20 0⎤⎦Q T ∈ S 3 + (17)This positive semidefinite matrix PX nearest X thus has rank 2, found bydiscarding all negative eigenvalues. The line connecting these two points is{X + (PX−X)t | t∈R} where t=0 ⇔ X and t=1 ⇔ PX . Becausethis line intersects the boundary of the positive semidefinite cone S 3 + atpoint PX and passes through its interior (by assumption), then the matrixcorresponding to an infinitesimally positive perturbation of t there shouldreside interior to the cone (rank 3). Indeed, for ε an arbitrarily small positiveconstant,⎡ ⎤λ 1 0X + (PX−X)t| t=1+ε= Q(Λ+(PΛ−Λ)(1+ε))Q T = Q⎣λ 2⎦Q T ∈ int S 3 +0 ελ 32.1.7.2 Tangential line intersection with boundary(18)A higher-dimensional boundary ∂ C of a convex Euclidean body C is simplya dimensionally larger set through which a line can pass when it does notintersect the body’s interior. Still, for example, a line existing in five ormore dimensions may pass tangentially (intersecting no point interior to C[161,15.3]) through a single point relatively interior to a three-dimensionalface on ∂ C . Let’s understand why by inductive reasoning.Figure 11(a) shows a vertical line-segment whose boundary comprisesits two endpoints. For a line to pass through the boundary tangentially(intersecting no point relatively interior to the line-segment), it must existin an ambient space of at least two dimensions. Otherwise, the line is confinedto the same one-dimensional space as the line-segment and must pass alongthe segment to reach the end points.Figure 11(b) illustrates a two-dimensional ellipsoid whose boundary isconstituted entirely by zero-dimensional faces. Again, a line must exist inat least two dimensions to tangentially pass through any single arbitrarilychosen point on the boundary (without intersecting the ellipsoid interior).